Simulation of heat sources
Mathematical description of turboset
Though this is not a topical question for the experts of TGSV, but the author found it necessary to include
it because he devoted to it a certain time of his life.
Wheelspace is divided into compartments – turbine sections with a constant flow rate. In the end of the
compartment there is an output to the next compartment, steam bleedings on regenerative heaters
(uncontrolled bleed), on production and central heating (controlled bleeds). In the end of a compartment
there is a steam-out through seals.
In terms of object-oriented simulation a compartment can be presented as an object consisting of the
Port 1. Inlet steam from adjacent compartment.
Port 2. Additional inlet steam into a compartment, for example, after the charging valve.
Port 3. Steam outlet into adjacent compartment
Port 4. Steam outlet through steals.
Port 5. Steam outlet on breeding.
Port 6. Mechanical power created by the steam.
Compartment also contains the following channels-characteristics:
o Compartment designation,
o Compartment diameter,
o Installation angle of blades,
o Rotor speed,
o Holdout flow rate of steam,
o Velocity coefficient,
o Degree of reaction of compartment,
o Peripheral speed,
o Fictitious speed,
o Speed ratio,
o Optimum speed ratio,
o Maximal relative efficiency ratio,
o Efficiency ratio,
o Isotropic enthalpy,
o Available enthalpy,
o Mechanical efficiency.
Vapour as the heating medium (port), consists of the following channels (parameters):
o mass flow rate,
o dryness factor,
o specific volume,
o number of area (zone) in I-S diagram (see section « Equations of water steam condition»).
Value of each channel (except for channels with a text, such as name) is determined by corresponding
formula, for example:
Diameter of a compartment (d). At calculation it is necessary to use average value of diameter of a
compartment. In reference literature we found out average diameters of the first (regulating) and the last
stage. Other diameters were accepted by the results of turbine setup on nominal conditions.
Angle of blade (1). At calculation angle of blade of 15° counted into the line target was used.
Rotor speed (n). Standard speed is about 50 rotations/s
Holdout flow rate of steam (Gо). Rated flow of steam through the compartment in adjusting mode was
taken as holdout flow rate of steam.
Velocity coefficient (). At calculation velocity coefficient was taken as 0,95.
Degree of reaction of turbine (о). At calculations the degree of reaction of turbine was taken as 0,05.
Peripheral speed (u)
Fictitious (virtual) speed (Cf)
Cf=(ho-hs) , (10.2)
ho- vapor enthalpy at compartment inlet,
hs- isentropic enthalpy after steam expansion in compartment.
Sped ratio (Y)
Y= u/cf. (10.3)
Optimum speed ratio (Yopt)
Yopt =*cos1/(2*(1-о) ). (10.4)
Speed ratio (J)
J= Y/ Yopt. (10.5)
Maximal relative efficiency ratio of (ηoimax)
For one-crown control stage 
ηoimax=0,83-(2E-4)*(P1/V1) /G1, (10.6)
For two-crown control stage
ηoimax=0,8-(2E-4)*(P1/V1) /G1, (10.7)
For other stages
ηoimax=0,925-(2E-4)*(P1/V1) /G1, (10.8)
P1- vapour pressure at compartment inlet,
V1- specific volume,
G1- vapour flow rate at compartment inlet.
ηoi = ηoimax*(2,1*J-1,19*J +0,09*J ), (10.9)
Isentropic enthalpy (hs)
GetHbyPaS – formula for enthalpy calculation by pressure and entropy (see appendix),
P2- pressure at compartment outlet,
S1- entropy at compartment inlet.
ho – vapour enthalpy at compartment inlet.
Vapour pressure at compartment inlet (the same pressure will be at vapour outlet from adjacent
compartment). Flugel formula  is used for determination of vapour pressure
2 2 2 2 1/2
P1=(P2 +(G1/G1op) *(P1op -P2op )*(T1/T1op)) , (10.13)
P1op – holdout vapour pressure at compartment inlet,
P2(P2op) – vapour pressure (holdout pressure) at compartment outlet,
G1 (G1op) - flow rate of vapour (holdout flow rate) at compartment inlet,
T1(T1op) – vapour temperature (reference temperature) at compartment inlet.
Vapour enthalpy at compartment outlet
h2= hs+(1- ηoi)*h. (10.14)
Dryness factor of vapour at compartment outlet
SteamX3byPaH(P;h) - formula for calculation of dryness factor by pressure and enthalpy (see
Vapour entropy at compartment outlet
GetSbyPaH(P;h)- formula for calculation of vapour entropy by pressure and enthalpy (see appendix).
Specific volume of vapour at the outlet
GetVbyPaH(P;h) - formula for calculation of specific volume of vapour by pressure and enthalpy (see
Area of finding a point of vapour state at compartment outlet (see section «Equations of water steam
GetZonePaT(P;T) - formula for calculation of vapour state area by pressure and temperature (see
Mechanical capacity of i compartment transferred by turbine shaft
Nmi=G1*(h1-h2)* ηmec,+Nmi-1, (10.18)
h1 – vapour enthalpy at compartment inlet,
h2 - vapour enthalpy at compartment outlet,
ηmec - mechanical efficiency of compartment,
Nmi-1- mechanical capacity transferred by turbine shaft from (i-1) compartment.
Validity of simulation results
Validity of simulation results can be estimated only by comparison of the results received in the
program MODEN with the results of experimental investigation. But at simulation it is necessary to repeat
boundary conditions of experimental works. MODEN allows to fulfill computer (computing) experiments,
the results of which can be compared to full-scale experiments (see [3,4,5]).
Experiment 1. Comparison of thermal characteristics of turbine PT-60/70-130 in condensation modes
Fig. 10.1. Comparison of experimental results
Heat pump stations
Refrigerators and recently heat pumps became an integral part of modern air conditioning systems.
Vapour compression refrigerators are really popular. In vapour compression refrigerators, for example,
freon with various additives is used as coolant. Simulation of such systems without sources of cold (heat)
does not allow making correct conclusion about economy operation modes, variation consideration of
consumer loads, time and values of climatic discomfort, etc. It is clear that it is mathematically
complicated to describe not only system, but also a separate element of refrigerator. We will try to present
such simplified model which can be quite sufficient for the experts of air conditioning systems.
Vapour compression refrigerators consist of the following basic elements (see fig. 11.2): evaporator
(03), condenser (01), compressor (02) and throttle (04). Evaporator is an equivalent to heat exchanger
where evaporation of liquid coolant and cooling-down of external heating medium takes place. Condenser
is a heat exchanger where condensation of coolant and heat of external heating medium takes place.
Difficulty in description of refrigerators becomes more complicated because in the directories and
catalogues machine data, received at standard operating conditions, is given. We cite a simple way of
recalculation of parameters by conditions which occur at real time-use.
Fig. 10.2. Function chart of heat pump
Equation of heat exchange in evaporator
Qi=(U*A) ev* (T1in-T1out)/(ln((T1in-Tev)/(T1out-Tev))) , (10.19)
Equation of heat exchange in condenser
Qcon=(U*A) con*(T2out-T2in)/(ln((Tcon-T2in)/(Tcon-T2out))), (10.20)
Qi – heat power of evaporator, W,
Qi - heat power of condenser, W,
T1in- temperature of heating medium evaporator inlet, °C,
T1out- temperature of heating medium at evaporator outlet, °C,
T2out- temperature of heating medium at condenser outlet, °C,
T2in- temperature of heating medium at condenser inlet, °C,
Tcon- temperature of condensation in condenser, °C,
Ti – temperature of evaporation in evaporator, °C,
(U*A) con - product U*A for condenser,
(U*A) ev – product U*A for evaporator.
Refrigerator is tested at standard conditions. If we assume that value (U*A) ev and (U*A) con do not
vary, then the system of equations for heat exchange will be as follows
Qsi=(U*A) ev* (T1in-T1out)*(ln((T1in-Tev)/(T1out-Tev)), (10.21)
Qscon=(U*A) con*(T2out-T2in)/Ln((Tcon-T2in)/(Tcon-T2out)), (10.22)
Qsi – standard heat power of evaporator, W,
Qsi – standard heat power of condenser, W,
T1in- standard temperature of heating medium at evaporator inlet, °C,
T1out- standard temperature of heating medium at evaporator outlet, °C,
T2out- standard temperature of heating medium at condenser outlet, °C,
T2in- standard temperature of heating medium at condenser inlet, °C,
Tcon- standard temperature of condensation in condenser, °C,
Ti – standard temperature of evaporation in evaporator, °C.
From the equations for standard conditions we can find the values of (U*A) ev and (U*A)con which then
are substituted into the equations for current values. Standard rating conditions of heat pumps are
prescribed, for example, by Eurovent. Some standard conditions are presented in the tables 10.1 and
Standard parameters of heat pump "water-water" testing
Parameters T1in T1out T2in T2out
Values 12 7 40 45
Standard parameters of heat pump "air-water" testing
Parameters T1in φ1in T1out φ1out T2in T2out
Values 12 60 7 87 40 45
Simulation has shown that transformation coefficient (COP) of heat pump does not fully depend on the
external temperature. It is connected with the fact that at all periods of the year we maintain constant
temperature in hydraulic mixer. Some fluctuations of COP are connected with temperature variation of
water in the container of coarse filter. Average values of COP from the start of heat pump to stationary
conditions are presented in the figure 10.3. As we can see from the figure, its value verges towards value
3.4. Therefore, in the following, we will consider that average transformation coefficient HP is CОР=3,4.
Achievement of the maximal values of transformation coefficient is one of the main problems at the
selection of heat pumps and schemes of their installation. Therefore at simulation we tried to select such
operation modes of TH work where COP was maximal. It is possible to sum up requirements which lead
to the growth of COP (for specified HP):
Temperature decrease of water inlet and outlet condenser,
Temperature increase of condensation (in condenser),
Temperature increase of evaporation (in evaporator),
Temperature increase of water inlet (outlet) in evaporator.
Fig. 10.3. Chart of COP variation from the start of heat pump to the stationary conditions
Operating mode of a heat pump
Heat pump has three basic operating modes:
Emergency switching mode.
There are only two modes in the model (the second and the third are incorporated). These modes are
determined by the value of number n which can take only two values 0 and 1. If HP is at normal mode,
then n=1 if it does not work n=0. The diagram of operating mode of a heat pump is presented in the figure
It is clear that operating mode of a heat pump depends on the external temperature. The higher
external temperature, the more time HP will be working.
So, at Тн =-20°C HP does not pass through shutdown condition, since its power is not enough for
water heating in hydraulic mixer to the upper limit (55°C).
Fig. 10.4. Operating mode of a heat pumpТн =-10°C
T-i diagram of a heat pump work
Basis for the analysis of HP work is T-i (T-s) work diagram.
Fig. 10.5. Т-I diagram of heat pump
Evaporator work of heat pump
As it was said earlier if you want to raise COP it is necessary to raise temperature of water inlet
(outlet) in evaporator. Temperature increase is possible only up to a certain limit (not higher than
temperature of water in the container of coarse filter). The maximal approximant to this temperature
causes the growth of a tube surface that should be put into the container (to raise a surface of heat
exchange). The decrease of this temperature leads to the decrease of COP and to danger of break by the
minimal temperature of water outlet from the evaporator.
Fig. 10.6. Diagram of temperature variation of heating mediums in evaporator.
Condenser work of heat pump
Fig. 10.7. Diagram of temperature variation of heating mediums in condenser
Water temperature in hydraulic mixer
The assumption that all water volume of heating systems is concentrated in hydraulic mixer was made
during the calculation. This volume is known, it is 13 m . Graph of temperature variation in the mixer from
start of a heat pump (Tout =-10°C) is presented in the figure 10.8. At start of HP this volume starts to
warm up gradually and in about 8 hours reaches 55°C (upper limit). HP stops. But heat analysis from
mixer proceeds and it cools down. Achieving 53°C (lower limit) HP activates. The temperature of water in
the tank starts to grow.
Fig. 10.8.Water temperature in a mixer
Fig. 10.9.Temperature in all inlet ports of hydraulic mixer
Horizontal ground heat exchangers
From the point of view of heating experts, ground is ever-living source of thermal energy. It is possible
to select geothermal heat (ground heat) only by means of heat pumps. Heat pump is a device which
allows transferring heat from cold (low-temperature) source to warm (high-temperature) consumer. Heat
pumps that are used for selection of ground heat are called sometimes as ground. This notion is rather
conventional since the same heat pump can be used both for selection of ground heat and for selection of
water and air heat.
At selection of the Earth heat top layers, placed at a depth of about 100 meters from the surface, are
used. From the point of view of heat exchange this layer is under influence of radiant energy of the Sun,
radiogenic heat from deep layers of the Earth, convection heat exchange with atmospheric air and heat
transfer at the expense of various mass transfer processes (rain, snow melting, ground water, etc.).
There are various classifications of the ground in foreign literature. Most of all we are interested in
classification of the ground by heat conduction. Data of famous American directory ASHRAE  is given
in the table 10.3.
Ground classification by 
Ground mode λ, Ground type
Very low heat <1 Light clay (15% moisture
Low heat conductivity <1,5 Heavy clay (5% moisture
Normal heat <2 Heavy clay (15% moisture
Light sand (15% moisture
High moisture content <2,5 Heavy sand (5% moisture
Very high heat >2.5 Heavy sand (15% moisture
We found in Russian sources tables 3 SNiP 2.02.04-88  on the basis of which it is possible to
create a table 10.4 by determination of heat conduction of a thawed ground - λth.
Ground classification by 
Ground mode λ, Ground type
Very low heat <1 Peaty soil and peat
Low heat conductivity <1,5 Loamy soil and clay, dust loamy
sand, light dust loamy sand
Normal heat conductivity <2 Heavy dust loamy sand
High moisture content <2,5 Heavy sand (5% moisture content)
Very high heat >2.5 Heavy sand (15% moisture content)
From comparison of the tables 10.3 and 10.4 is clear that data of American and Russian directories is
rather adequate. For exact determination of heat conduction of the grounds, it is necessary to conduct
experimental researches of heat conduction in a place of prospecting construction activities.
Let us note that heat conduction of a ground is not a constant value within a year. It depends on
humidity, aggregative state of moisture in a ground and temperatures. And most of all humidity varies at a
ground freezing. Data  shows that heat conduction of frozen grounds λ f is
λf=1.05…2.1* λth. (10.23)
The author has only foreign sources (see fig.10.10 ) of data about ground temperature at various
depths. From these data it is possible to draw a conclusion, that at a depth of more than 8 meters the
temperature is practically constant within a year (only 1/20 changes on a surface). There is such notion
abroad as ground temperature. The directory ASHRAE  offers to determine ground temperature by the
temperature of ground waters in the relevant locality. If we take into account temperature of ground
waters, then it ranges from 8-10°C for conditions of Belarus.
Value of quantity of radiogenic heat is (for Central Europe) 0,05-0,12 W/m2 . If it is unknown, then
usually 0,1 Вт/m2 is taken.
Fig. 10.10Temperature distribution of a ground by depth.
There are two basic ways of geothermal heat selection - by means of open and closed circuits. By
open circuit we mean the use of heat of ground waters that specify delivery of these waters onto the
surface, the use of their heat and returning into the formation. By closed circuit we mean the use of heat
of a ground by means of intermediate heat exchangers and heating mediums. In turn systems with closed
circuits differ in the type of heat exchangers - horizontal (fig. 10.11а) and vertical (fig.10.11в). The device
of closed circuits with vertical heat exchangers is more expensive, than that one with horizontal heat
exchangers. At the same time circuits with horizontal heat exchangers occupy large territories that can
be, in some cases, a rather critical condition.
Fig. 10.11. Pipe distribution of horizontal (А), vertical (B) ground heat exchangers and variations of
horizontal heat exchanger placement into the pipe trench (C) 
Pipes of horizontal heat exchangers are placed into the pipe trenches. Placement of the pipes into the
trenches is usually carried out by two basic ways: straight line and spiral pipes. In reality there are also
other, sometimes rather exotic, ways, for example, pipelines are covered on top by copperplates (copper
fins) - probably for improvement of heat exchange.
In this article we could not take into consideration such question, heaving of heat pumps. Let us
consider some important characteristics of heat pumps: heat conduction, COP – transformation
coefficient, temperature of the heating medium at condenser inlet (or condensation temperature, which is
above 10°C) and antifreeze temperature at evaporator outlet. For standard conditions all these values are
given by the manufacturer of heat pump. We will choose well-known French firm CIAT. Data from the
directory for heat pump LGN-100Z of this firm is presented in the table. Heat pump LGN-100Z uses
R407c as coolant. We can easily get the value of COP from this table and it is
Diagrams of transformation coefficient dependence on the temperature of water (antifreeze) at
evaporator outlet are given in the figure 10.12.
Characteristics of heat pump LGN 100Z
(at condensation temperature of 55°C)
Water temperature at Qc, kW N, kW Qh, kW COP
the output from
Water- -8 13,8 9 22,8 2,53
glycolic -4 16 9,1 25,1 2,76
2 20,4 9,4 29,9 3,18
Water 5 23,2 9,5 32,7 3,44
7 25 9,5 34,6 3,64
12 30 9,6 39,6 4,16
The choice of heat pump at design stage is not a simple problem. The reason is that heat pump is
never selected for total peak heating load. If it happens, then capital expenditure will be so large, that
payback of your decision will not ever happen. It is clear, that if you choose heat pump not for peak load,
it will be necessary to find special peak terminal unit. Electric boilers are usually applied as peak terminal
units. And again a question arises: « What load should be used for heat pumps? ».
Fig.10.12. Characteristic chart of the heat pump - LGP.
Work simulation of a heat pump together with a ground heat exchanger can be held by means of the
simulation program - MODEN, developed by Energovent . On the basis of numerous computer
experiments and comparisons of the received data with known foreign methods, and also taking into
consideration the experience of the heat pump device with ground heat exchangers on water abstraction
Muhavetskiy (Brest), the recommendations on designing such systems were prepared in Energovent. In
this article we will take only some data from these recommendations.
For demonstration of this fact let us look at the behaviour of horizontal heat exchanger that consists of
two pipes Dнар=32 mm placed in a trench 100m in length. Initial temperature of a ground will be 10°C
and antifreeze temperature will be minus 10°C and we select heat of a ground (heat conduction of a
ground is 3 /(m*°C)
One more question. What is the power of a ground as a heat exchanger during heat pump work? For
answering this question we will conduct a computer experiment.
In the figure 10.13 there are results of the computer experiment executed within the limits of the program
MODEN (version 2.1). The experiment begins on the 1 of November, and ends on the 1
Fig. 10.13.Results of computer experiment in MODEN
Let us analyse the graph. At the first stage it is possible to select more than 4000 W, but this period
lasts not long, then there is a fall up to 1500 W, and then a small growth starts again. Falling of a heat
extraction is connected with a ground cooling (heat exchanger work and pulldown of temperature of
external air), and the increase is caused by the increase of solar radiation and some increase of
temperatures of external air. Average heat extraction for settlement period will be QCр=2232 W. Actually
we have this heat extraction. At some period it can be increased, and vice verse, but average heat
extraction should not exceed this figures. Let us take rather simple technique to find out the value of the
required heat pump capacity that is very widespread in the USA.
We determine the number of hours when the maximum of thermal capacity is used
Kзагр=max /y. (10.26)
Rated capacity of the heat pump in foreign literature is offered to determine by the formula
QTout=2*Qmax* Kзагр. (10.27)
Coefficient 2 in this formula takes into account that duration of the heating period is, approximately, 50 %
of all the year. Now we will not prejudice this formula, though, undoubtedly, it requires specification.
The capacity of ground heat exchanger is determined by simple formula
As it was before, pipelines of horizontal heat exchanger are placed into the trenches. The number of
pipes in the trench can be different (1, 2, 4, 6, 10, etc.), as well as the distance between trenches. What if
we put all the pipes into one trench? Allure of such assumption is erroneous because both between close
pipes, and between trenches interference can happen, i.e. imposition of temperature fields, that leads to
essential decrease of a heat flow from the ground to pipes. Therefore, if we assume that the number of
pipes placed into a single trench with one pipe is Lтр1, then the number of pipes in reality is
Kтр – coefficient of correction that takes into account the number of pipes in a trench (see tab. 10.6),
Kтран – coefficient of correction that takes into account the distance between trenches. Calculations
show that if the distance between trenches is more than 2 m, then Kтран=1.
Lp=F(Kтр, Kтран, Tгр, Тж, λth, Cгр, Dнар, λCт….) . (10.30)
It is not always possible to take into consideration all the factors by introduction of coefficients (Kтр
and Kтран). Direct calculation for certain conditions is the most preferable. Nowadays such direct
checking calculation can be executed by means of simulation programs. Making a number of such
calculations, we want to show the reader of the article how separate parameters influence the value of
heat extraction by heat exchanger.
Influence of a ground temperature, heat conduction of a ground and antifreeze temperature (heating
medium circulating through the evaporator of heat pump) can be estimated by means of the table 10.7.
We can see from the table that the temperature of heating medium influences the value of heat extraction
strongly then goes heat conduction of a ground and its temperature which influences this value most of
all. Though this data is received by means of numerical calculation, the author does not understand
exactly why the influence of thermal conductance coefficient is so insignificant? I thought that it will be a
direct proportion, as it was in case of a stationary problem of heat conduction. In foreign sources that I
read this dependence is not discussed at a numerical level.
Correction coefficient by the number of pipes in one trench - Ктр
Base Numbers of tubes in a
2 4 6
Calculation in MODEN 1,45 1,97 2,34
 1,43 1,73 2,16
Heat transfer of horizontal ground heat exchanger
from 2 pipes and the trench length of 100 m
Тгр, °C λth, W/(m*°C) λth, W/(m*°C) λth, W/(m*°C)
at Тж=-10 °C at Тж=-6 °C at Тж=-2 °C
1,5 3 1,5 3 1,5 3
8 1839 2287 1307 1561 794 833
10 1951 2391 1428 1674 897 958
Example. Calculate horizontal ground heat exchanger and choose heat pump for a building heating
located in Minsk. Design load on the heating system is 80 kW, heat conduction of a ground is W/(m*°C)
1. For Minsk at heating system work at external temperature below +8°C, the value of maximum hours is
2320 (it is calculated in the program MODEN), thus charge coefficient is 0,264. We determine required
capacity of heat pumps by the formula (10.27)
2. So as to accept the most effective variant of installation, we make a number of calculations on various
values of the number of pipes in a trench (1, 2 and 4) and antifreeze temperature (-10,-6 and-2°C). We
show how to make one of the calculations: one pipe in a trench and antifreeze temperature -10°C
3. We install a heat pump of the firm CIAT and of the LGN mark. According to the graph in the fig. 10.12
for such heat pumps we choose COP that is equal 2.35.
According to the formula (10.28) the capacity of a ground heat exchanger will be
Qгт= 63,3*(2,35-1)/2,35=36,36 kW.
4. According to the table 10.7 there is no data on heat transfer in the trench with a single pipe. Therefore
we take a similar trench, but with 2 pipes. Average value of the selected heat from the trench of 100 m for
the heating period is 2391 W. On 100 m of the pipe heat removal will be
5. If there is only one pipe in a trench instead of two pipes, then heat removal subject to Ктр will be
6. Total length of pipes and trenches will be
7. Flow rate of electric energy is determined by the formula
8. The analysis of calculation results shows (calculations are not presented) that the most economical,
according to the capital expenditure, are variants with antifreeze temperature of -10°C. These are the
variants with a minimum number of pipes and a greater standard size of the heat pump. Such projects are
more popular in the Central Europe that is connected with a lack of space. In spite of great capital
expenditure, the variant with antifreeze temperature of -2°C has smaller operating costs that is connected
with a smaller size of the heat pump. A great number of pipes require a lot of space. Such projects are
more popular in the USA and Canada.
It is impossible to say what variant has real advantages. Deviations between the expenses are within the
bounds of computing experiment.
1. In the article we considered major factors influencing the work of horizontal ground heat exchangers:
heat conduction of a ground, interaction of pipes in a trench and interaction of tranches among
themselves, antifreeze temperature in a circuit of heat exchangers.
2. In the article we presented fundamentals of design technique of heat pump systems with horizontal
ground heat exchangers. This technique is made on the basis of computer experiments executed by
means of the program MODEN (version 2.1).
3. There is an example of heat extraction of heat pump system using the received results.
COP- coefficient of heat pump transformation,
Сгр – specific heat of a ground, J/(m *°C),
Dнар – external diameter of pipeline of ground heat exchanger, m,
Kзагр – charge coefficient of heat pump,
Kтран – coefficient that takes into account distance between trenches,
Kтр – coefficient that takes into account the number of pipes in a trench,
Lтран – length of a trench for allocation of horizontal ground heat exchanger, m,
Lтр – length of the pipelines of horizontal ground heat exchanger, m,
N – electric power of heat pump drive, W,
Тгр – ground temperature, °C,
Тж – average antifreeze temperature (of heat exchanger passing through evaporator),°С,
Qгод - total annual expenditure of thermal energy, J,
Qmax - maximum heat load, W,
Qh – heat output of HP, W,
Qc - cooling efficiency of HP, W,
Qтн – required heat output of HP, W,
Qгт – heat power of ground heat exchanger, W,
max - number of hours used for maximum of heat load, s (hour),
год – year duration, (hour),
λ – heat conduction, W/(m*°C),
λth - heat conduction of thawed ground, W/(m*°C),
λf - heat conduction of frozen ground, W/(m*°C),
λст - heat conduction of pipe wall, W/(m*°C).
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