- Email: [email protected]
Method of Analytical Calculation Radiating (Steam) Superheater of the Steam Locomotive
Proceedings of the 13th IFAC Symposium on Information Control Problems in Manufacturing Moscow, Russia, June 3-5, 2009
Method of Analytical Calculation Radiating (Steam) Superheater of the Steam Locomotive Е.А. Olenev Vladimir state university, Vladimir, Russia Abstract: The analytical method of calculation radiating superheater is offered, allowing defining temperature of overheated steam in the course of steam locomotive movement. This method can be used for calculation and other heat exchangers.
1. INTRODUCTION Application on a railway transportation of the steam locomotives using cheap fuel – Coal, will allow to lower expenses for transportation of cargoes and passengers. Thus the steam locomotive, being more economic in comparison with diesel locomotives and electric locomotives which work on secondary energy carriers, should be free now from the basic lacks – low EFFICIENCY and heavy conditions of operation. One of ways of increase of EFFICIENCY of steam power plant, as is known, is application overheat steam. Applied on old steam locomotives a convection superheater could not provide a heat of an overheat of steam as they were shielded by a cold surface fire tube through which heat was transferred to boiler water. To raise temperature of an overheat of steam in a steam locomotive it is possible by means of radiating superheater in which heat from combustion products is transferred by radiation and convection. Thus as fuel it is better to use the gas energy received by thermal decomposition of firm fuel directly in a steam locomotive fire chamber (Olenev 2007, 2008). Methods of calculation radiating superheater, resulted in the literature are grafo-analytical, containing subjective estimation elements at a choice and appointment of some settlement parameters that does not allow constructing the mathematical model, giving the chance to spend necessary calculations in real time in the course of automatic control. For the purpose of elimination of the specified lacks, the method has been developed and described more low, which can be applied to calculation and modeling of work radiating superheater and other heat exchangers. 2. DEFINITION OF COEFFICIENT OF A HEAT TRANSFER OF SUPERHEATER
The coefficient of heat emission can be found by radiation under the known formula
αr =
4 ε g C ⎡⎛ Tg ⎞ ⎛ Tw ⎞ 4 ⎤ ⎢⎜ ⎟ −⎜ ⎟ ⎥=
Tg − Tw ⎢⎜⎝ 100 ⎟⎠ ⎣
⎝ 100 ⎠ ⎥ ⎦
2 2⎤ ⎡ ⎛ Tg T ⎞ ⎛ Tg ⎞ ⎛ Tw ⎞ ⎥ ⎟ +⎜ = 10 −2 ε g C ⎜⎜ + w ⎟⎟ ⎢⎜⎜ ⎟ ⎟ ⎝ 100 100 ⎠ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦
,
(1)
where α r – coefficient of heat emission by radiation, V/(m2·°С); coefficient;
С = σ ⋅108 = 5,67 W/(m2·°K4) – emission σ – a radiation constant, equal
5,67 ⋅ 10 −8 W/(m 2 ⋅ K 4 ) ; ε g – degree of blackness of gas; Т g , Т w – accordingly absolute temperature of radiated gas
and a heat exchanger wall (an element of superheater), °K. Through unit of a surface of the heat exchanger, having on the one hand temperature of smoke gases t g , and with another – temperature of the heated up environment t s (°С), the following thermal stream will be transferred
(
)
q = K t g − ts ,
(2)
where q – heat stream through surface unit, W/m2; K – heat transfer coefficient, W / (m2 °С); t s – temperature of the heated up environment. Same most it is possible to receive, using coefficients heat emission on both parties heat exchanging surfaces
Let us present radiating superheater in a kind of heatinsulated pipes, in which (steam) super heater elements are placed, washed by burning products, moving a countercurrent. Heat exchange process between a wall (steam) super heater element and smoke gases washing it, grows out of joint action of radiation, convection and heat conductivity. The quantitative characteristic of the specified process is the coefficient of heat emission α g = α r + α с ,
978-3-902661-43-2/09/$20.00 © 2009 IFAC
where α r considers influence of emission, and α с – convection and heat conductivity.
′ , q = α g t g − tw
(
)
(3)
′′ − ts ) , q = α s (t w
(4)
′ , t w′′ – accordingly temperature of a wall of the heat where t w exchanger from smoke gases and the heated up environment, °С; α s – coefficient of heat emission on the party of the heated up environment, W / (m2 °С).
1708
10.3182/20090603-3-RU-2001.0507
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
As in metal heat exchangers temperature drop in a wall makes very small size (Several degrees), the thermal resistance of a wall, can be neglected, and ′ = t w′′ = t w . Then, equating the right parts of the consider t w equations (3), (4), we will receive
Summarize the equations (8) and (9), we will receive the expression describing degree of blackness of gas, containing simultaneously carbonic acid and water steam
That can be neglected thermal resistance of a wall tw =
α g t g + α sts . αg +αs
(5)
For calculation the coefficient of heat emission radiation, we will accept more simple dependence of temperature of a wall on temperatures of smoke gases and environment tw =
t g − ts . ln t g t s
(
)
(6)
Taking into account expression (6) and some transformations the equation (1) will become ⎛ ⎞ t g − ts α r =10 −8 ε g C ⎜ Tg + + 273 ⎟ × ⎟ ⎜ ln t g t s ⎝ ⎠
(
)
2 ⎡ ⎛ ⎞ ⎤ 2 ⎜ t g − ts ⎢ ⎟ × Tg + + 273 ⎥ ⎜ ln t g t s ⎟ ⎥ ⎢ ⎝ ⎠ ⎥⎦ ⎢⎣
(
.
(7)
)
As is known, considerable radiating and absorption abilities, having practical value, multinuclear gases possess, in particular, СО2 and Н2О. Unlike solids, the radiation and energy absorption in gases occurs in volume, and their radiating and absorption abilities depend on gas temperature t g , lengths l ways of a beam and partial pressure р. For values pl from 0,006 to 1atm and temperatures from 700 to 2000°С by the author had been received formulas, defining relative radiating ability (or blackness degree) carbonic gas and water steam: ⎡5 ⋅10 −4 ⋅ t g − 3,1 +
ε СО2 = 10
−2 ⎢
ε H 2O = 10
−2 ⎢
(
⎢+ 32,987 − 9,82 ⋅10 −3 t g ⎣
⎡5 ⋅10 −4 ⋅ t g − 3,1 +
(
⎢+ 48,83 − 1,496 ⋅10 −2 t g ⎣
⎤ ⎥, 0, 2788 ⎥ pCO2 l ⎦
(8)
⎤ ⎥ pH 2Ol 0,4313 ⎥⎦
(9)
)( )(
)
)
Where ε СО 2 , ε H 2О – accordingly degree of blackness of carbonic acid and water steam; pCO2 , pH 2O – accordingly partial pressure of carbonic gas and water steam, atm; l – length of a way of a beam, m;
(
)(
)
0, 2788 ⎤ ⎡t − 6200 + 32987 − 9,82t p + g g CO 2 l ⎥. 0, 4313 ⎢+ 48830 − 14,96t p ⎥ g H 2O l ⎣ ⎦
ε g = 10 −5 ⎢
(
)(
)
(10)
Total radiation of such mix is a little bit less than sum of radiations of carbonic acid and water steam, containing in it, as strips of radiation and absorption for СО2 and Н2О partially coincide, leading to mutual partial absorption of energy radiated by them. However at usual parities of components, taking place in smoke gases, the amendment on radiation decrease, possible not to consider. The equation for coefficient of heat emission by radiation will take on form ⎛ ⎜ ⎝
α r = 10 −13 C ⎜ Tg +
⎞ + 273 ⎟ × ⎟ ts ⎠
t g − ts
(
ln t g
)
2 ⎡ ⎛ ⎞ ⎤ 2 ⎜ t g − ts ⎢ × Tg + + 273 ⎟ ⎥ × ⎜ ln t g t s ⎟ ⎥ ⎢ ⎝ ⎠ ⎦⎥ ⎣⎢
(
)
(
)(
⎡t − 6200 + 32987 − 9,82t p g g CO 2 l ×⎢ 0, 4313 ⎢+ 48830 − 14,96t p g H 2O l ⎣
(
)(
)
)0,2788 +⎤⎥ ⎥ ⎦
.
(11)
The coefficient of heat emission by contact is defined basically by conditions of movement and physical properties of gas. At the established turbulent movement of a stream the coefficient of heat emission by contact can be calculated on both parties of a heat-transmitting surface under the known formula
α с = 1,163B
(ωρ )0,8 d 0, 2
= 1,26 ⋅ 10 −3 В
(Vρ )0,8 S 0,2 , A
(12)
where α с – coefficient of heat emission by contact, W/(m2·°С); В – correction multiplier; ω – speed of moving gas, m/s; V – quantity of moving gas, nm3/h; ρ – gas density, kg/nm3, d – the resulted diameter of the channel on which gas moves, m; S – perimeter of the channel on which there is a gas, m; A – live section of the channel, m2. Value of multiplier B calculated from a following parity В = 0,313
λ0,6с р 0,4 η 0, 4
,
(13)
where λ – heat conductivity of gas, W / (m °C); с р – a gas thermal capacity at constant pressure, kJ / (kg °C); η – dynamic viscosity, Pa·s. On the basis of expressions (12), (13) we can write down
1709
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
α с = 3,95 ⋅10 −4
λ0,6с р 0,4 (Vρ )0,8 0,2 S . A η 0, 4
η а = 15,5 ⋅ 10 −6 + 0,28 ⋅ 10 −6 t a 0,69 .
(14)
For smoke gases at humidity 5-20 % in the range of temperatures from 0 to 1500°С have been received following dependences of coefficients of a thermal capacity c pg , kJ / (kg °C), heat conductivity λ g , W / (m °C), and viscosity
η g , Pa·s, gases from their temperature. c pg = 1,032 + 3,13 ⋅10 −4 t g 0,865 ;
λ g = 2,21⋅10 −2 + 4,65 ⋅10 −5 ⋅ W +
(
)
+ 7,56 ⋅10 −5 + 6,98 ⋅10 −7 ⋅ W t g
;
α сu =
(17)
where W – humidity content of gas, %. At pressure of water steam from 12 to 20 kg/sm2, characteristic for locomotive boilers, in the range of temperatures 240 ÷ 500 °C thermal capacity c pu the pair can be described the formula с рu =
16,34е
0,094 рu
+ 1,957tu + 66,2 , tu
(18)
kg/sm ; tu – steam temperature, °С. Heat conductivity λ u and dynamic viscosity ηu of steam in the specified range of pressure and temperatures from 200 to 700 °С will accordingly be expressed
ηu = 8 ⋅ 10 −7
(tu + 273)1,632 .
c pa = 0,9897 + 10 t a + 3,243 ⋅ 5,8
.
(22)
Heat conductivity of air at temperature from 0 to 1500 °С
λа = 2,44 ⋅10
−2
−4
+ 1,45 ⋅10 ta
0,865
,
0, 4
(tu + 273)0,5832 . (tu + 946)0,2
(Vρ )0,8 S 0,2 (168,3+t A
a
)
0,865 0,6
⎛ 0,9897 + 10 − 4 t + 3,243 ⋅5,8 −(3+0,004ta ) ⎞ a ⎟ ×⎜ −6 −6 0,69 ⎟ ⎜ t 15 , 5 10 0 , 28 10 ⋅ + ⋅ a ⎠ ⎝
(26)
×
(27)
0, 4
.
)
(28)
where qs – quantity of heat received by the environment, kJ/h; Vs – quantity of environment, the past through the heat exchanger, m3/h; c eps , t se and c bps t sb – accordingly a thermal capacity, kJ / (kg °C), and temperature, °С, environments on an exit and input of a heat exchanger; ρ s – density of environment, kg/m3.
and
dynamic viscosity η a of air at atmospheric pressure in the range of temperatures from 0 to 1500 °С can be expressed as follows − (3+ 0,004ta )
3,122⋅105 A
α сa =1,965⋅10 −6
qg =
The thermal capacity c pa , heat conductivity λa
−4
(Vρ )0,8 S 0,2 ×
1,665 ⎤ ⎡ 5 +⎥ ⎢4,578⋅10 + 550 −tu ×⎢ ⎥ ⎢ + 65⎛⎜ pu (1000 −tu )+tu ⎞⎟ ⎥ ⎢⎣ ⎠ ⎥⎦ ⎝ 10
(21)
tu + 946
)
If through γ to designate a share of heat lost through walls radiating of superheater, the quantity of heat which smoke gases should give, will turn out equal
(20)
tu + 946
0, 4 (25) ⎛ 3297 +t 0,865 ⎞ g ⎟ ⎜ 61,25+t 0,724 ⎟ g ⎝ ⎠
0, 6 ⎜
qs = Vs ρ s c eps t se − c bps t sb ,
2
(tu + 273)2,06 ,
A
(
(19)
where ðu – pressure of steam in superheater elements,
λu = 1,088 ⋅10 −4
(Vρ )0,8 S 0,2 ×
Let's establish now interrelation between target temperatures of smoke gases and the heated up environment. The quantity of heat received by the environment in the heat exchanger will be equal
and in a range from 400 to 700 °С ⎡ 1,665 ⎤ +⎥ 4,578 ⋅105 + 550 − tu , c pu = 4,19 ⋅10 −6 ⎢ ⎢+ 65(0,1 p (1000 − t ) + t ) ⎥ u u u ⎣ ⎦
α сg = 1,408⋅10 −6
(
(16)
η g = 14,7 ⋅10 −6 + 0,24 ⋅10 −6 t g 0,724 ,
On the basis of the resulted expressions we will receive following equations for coefficients of heat emission by contact for smoke gases α cg , steam α cu and air α ca
× 31661,9 + 66,62W + (108,3 + W )t g
(15)
(24)
(
Vs ρ s c eps t x − c bps t sb 1− γ
) = V ρ (c g g
b b pg t g
− c epg t ge
)
(29)
where q g – the quantity of heat given by smoke gases, kJ / h; Vg – quantity of smoke gases, the past through the heat
exchanger, m3/h; c bpg , t gb and c epg , t ge ; – accordingly a thermal capacity, kJ / (kg °C), and temperature, °С, smoke gases on an input and exit of a heat exchanger; ρ g – density of gases, kg/m3.
(23)
1710
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
In the range of temperatures from 100 to 2000 °С for average structure of smoke gases, it is possible to express dependence of their thermal capacity on temperature
αs =
c pg = 1,038 + 1,12 ⋅10 −4 t g .
Hence, the coefficient of a heat transfer will become
(30)
On the basis of expressions (29) and (30) we will write down
(
)
2⎞ ⎛ 2 1,12 ⋅10 −4 ⎜ t ge − t gb ⎟ + 1,038 t ge − t gb + ⎝ ⎠
+
(
Vs ρ s c eps t se − c bps t sb Vg ρ g (1 − γ )
)=0
ts =
Vs ρ s
(1-γ )Vg ρ g
(c
e e ps t s
)
2
− c bps t sb + 1,12 ⋅10 − 4 t gb + 1,038t gb −
− 4634.
In a radicand, in subtracted, instead of a thermal capacity
c eps it is necessary to substitute function of its dependence on temperature, i.e. for steam it is the equations (18) or (19), and in case of heating, for example air – (22). After substitution we receive value of temperature of smoke
t se
t se
environment, i.e. being set by value , on the equation (32) we calculate the corresponding temperature of smoke gases on an exit. Having established, thus, interrelation between temperatures on an input and exit of a heat exchanger, using the equations (25), (26), (27), it is possible to define values α c for the concrete environment and α r from expression (11). After that for smoke gases the coefficients of heat emission α g is calculated. To true values of temperatures t se and
t ge
will correspond such coefficients of heat emission on
both party of a heat-transmitting surface which being are substituted in expression of coefficient of a heat transfer К, will provide equality
(
3,6 ⋅ К ⋅ Аw ⋅ t s − Vs ρ s c eps t se
− c bps t sb
)= 0 ,
(33)
where t s – average value of temperature of environment in the heat exchanger, °С; Аw – the area of a surface of heat exchange, m2. Let’s accept change of coefficients of heat emission along the heat exchanger under the logarithmic law then the average coefficient of heat emission is equal
αg =
α gb
− α ge ln α gb / α ge
(
)
,
(34)
e s
− α sb
(35)
(α − α )(α − α ) )ln(α / α )+ (α − α )ln(α b g b g
e g
e s
e g
b s
b g
e g
t se − t sb
(
ln t se / t sb
e s
/ α sb
),
).
(36)
(37)
(
)(
3,6 ⋅ Аw α gb − α ge α se − α sb
α se
(32)
gases on an exit from the heat exchanger depending on
(α
)
Substituting expressions (36), (37) in the equation (33), we will receive
(
= 94,5 ×
× 2405 −
(
We accept change of temperature of environment in the heat exchanger under the logarithmic law. Then
(31)
The reference temperature of smoke gases and environment, as a rule, is known, therefore, solving the equation (31) concerning temperature of smoke gases on a heat exchanger exit, we will have t ge
К=
α se − α sb . ln α se / α sb
−α sb
)(
ln α gb /α ge
)(
+ α gb −α ge
)
)ln(α
(
e b s /αs
)×
)
te − tb × s e sb − Vs ρ s c eps t se −c bps t sb = 0 . ln t s / t s
( )
(38)
Using dependences (15), (18), (19) and (22) thermal capacities of the heated up environment from temperature, having substituted this dependence instead of a parity
(c
e e ps t s
)
− c bps t sb and being set by value t se from the equation
(32) find t ge , then calculate on the equations (25), (26), (27), (11) values α s , α r and then α g to begin with and end of the heat exchanger. Then check justice of expression (38) which allows to judge true value of temperature t se . The equation (38) gives the chance to receive also value of temperature replace all
t se directly. For this purpose it is necessary to variables with their functions. For example,
instead of α gb to substitute expression from the equations (11), (25) ⎞ ⎛ t gb − t se ⎟× 10−13 C ⎜ t gb + 273+ + 273 b e ⎟ ⎜ ln t g t s ⎠ ⎝ 2⎤ ⎡ b e ⎞ 2 ⎛ t g − ts ⎢ ⎟ ⎥× × ⎢ t gb + 273 + ⎜ + 273 ⎜ ln t b t e ⎟ ⎥ g s ⎝ ⎠ ⎥⎦ ⎢⎣
(
(
)
(
(
)
)
)(
⎡t b − 6200 + 32987 − 9,82t b p g g CO 2 l ×⎢ 0, 4313 ⎢+ 48830 − 14,96t b p H 2O l g ⎣
(
+ 1,408⋅10−6
1711
)(
)
)0,2788 +⎤⎥ ⎥ ⎦
(Vρ )0,8 S 0,2 ⎛⎜ 31661,9 + 66,62W + ⎞⎟ ⎜ + (108,3 + W )t gb ⎟ A ⎝ ⎠
+ 0,6
×
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
⎛ b 0,865 ⎞ ⎜ 3297 + t g ⎟ ×⎜ ⎟ 0 , 724 ⎜ 61,25 + t gb ⎟ ⎝ ⎠
equation (36), W / (m2 °C). The temperature of a heattransmitting wall according to expression (5) in the beginning
0, 4
.
(39)
superheater will make
t wb = 843,7 °С, in the end –
and t s on etc. Thus Having replaced thus t g on bulky enough equation having, however, only one unknown
t we = 511,4 °С. The average temperature of steam in superheater according to the equation (37) will be equal t s = 530,3 °С
size tse which dares a dichotomy method will turn out. Such way is simpler and is comprehensible for the COMPUTER.
4. AN EXAMPLE 2. DEFINITION OF TEMPERATURE OF HEATING OF AIR
Having received value of temperature t se , then define all other necessary parameters.
To define temperature of air leaving the heat exchanger
tse ,
t gb
3. AN EXAMPLE 1. DEFINITION OF PARAMETERS OF THE OVERHEAT STEAM To define temperature leaving of superheater steam having reference temperature tub = 437 °С and pressure of pu = 16 kg/sm2. The expense of steam of Vu ρu = 4000 kg/h, quantity passing through superheater smoke gases Vg = 2000 m3/h. The share of heat lost through walls radiating superheater, makes γ = 5 %. Smoke gases contain 19 % СО2, 1 % Н2О, and 5 % of moisture and are included into the heat exchanger with temperature t gb = 1400 °С. Superheater is executed in the form of the cylinder in diameter of 1,1m in which are placed 80 tubes in the size of 24/30 mm for steam passage, having perimeter of Su = 6,03 m and live section Au = 0,0362 m2. Channel perimeter, on which движутся smoke gases, S g = 11 m, and its live section Ag = 0,8938 m2. Surface of heating Аw = 18,9 m2.
The decision of the equation (38) by method of a dichotomy, gives size of temperature of the leaving steam from the superheater tuе = 636,1 °С. Substituting the received value in the equation (32), we find temperature of smoke gases on an exit t ge = 874 °С. By means of expressions (25) and (26) we calculate coefficients of heat emission by contact to begin with and end the heat exchanger (For smoke gases and steam) = 6,16
е α cu
2
W/(m ·°С),
2
= 230,46 W/(m ·°С),
b α cu = 206,26 W/(m2·°С). Under the formula (11) it is had
α rb α gb
= 79,85
2
by smoke gases with reference temperature t gb = 1050 °С, containing 19% СО2, 1 % Н2О and 5 % of a moisture. Quantity of warmed up air Vа = 2500 m3/h. Volume of the smoke gases which are passing through the heat exchanger Vg = 2900 m3/h. Smoke gases moving in the cylinder in diameter of 1,1m, having perimeter S g = 3,456 m, and live section Ag = 0,95 m2. Outside of the cylinder the ring channel in width of 19 mm for air passage is placed. In the air channel are established 24 directing edges, by 14 × 6 mm section. Considering equal 6 mm a thickness of metal of the cylinder, we receive the internal diameter of the ring channel equal of 1112 mm and external diameter, equal 1150 mm. From here perimeter of the air channel of S а = 7,78 m and live section Aа = 0,0655 m2. The share of heat lost through walls of the heat exchanger, makes γ = 15 %. The area of a heat-transmitting surface Аw = 12,9 m2. The effective thickness of a radiating layer, product pCO2 l , pH 2Ol will be same, as well as in the previous example.
The effective thickness of a radiating layer for the cylinder in diameter of 1,1m will be equal to l = 0,9 ⋅1,1 = 1 m, therefore pCO 2 l = 0,19 atm, pH 2Ol = 0,01 atm.
b α cg
having reference temperature tаb = 420 °С. Air is warmed up
W/(m ·°С),
α re 2
2
= 37,03 W/(m ·°С).
= 79,85 + 6,16 = 86,01 W/(m ·°С),
α ge
Then
= 37,03 + 5,33
= 42,36 W/(m2·°С). According to expressions (34), (35) average coefficients of heat emission for smoke gases will be equal α g = 61,63 W/(m2·°С), for overheat steam W / (m2 °C). The heat transfer coefficient is defined from the
Solving the equation (38) we find temperature leaving the heat exchanger of air tае = 608,77 °С. Substituting this value in the equation (32), we find t ge = 889,8 °С. By means of expressions (25) and (27) we calculate coefficients of heat emission by contact to begin with and end b e = 5,65 W / (m2 °C), α cg = 5,38 W / heat exchanger α cg b e (m2 °C); α ca = 61,32 W / (m2 °C), α ca = 64,93 W / (м2 °C).
Under the formula (11) it is had α rb = 54,47 W / (m2 °C),
α re = 37,43 W / (m2 · °C). Then α gb = 54,47 + 5,65 = 60,12 W / (m2 °C),
α ge = 37,43 + 5,38 = 42,81 W / (m2 °C). According to expressions (34), (35) average coefficients of heat emission for smoke gases will be equal α g = 50,98 W / (m2 °C), for air α a = 63,1 W / (m2 °C). According to expression (36), K = 28,12 W / (m2 °C). The temperature of
1712
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
a heat-transmitting wall in the heat exchanger beginning will b make t w = 820,9 °С, in the end – t we = 613,2 °С. From the equation (37) average temperature of air will be equal t s = 508,6 °С.
5. CONCLUSION The stated method includes performance of sequence of formal operations, owing to what completely all calculation of the heat exchanger can be carried out on the computer. The received mathematical model, gives the chance to spend in real time necessary calculations for automatic control of boiler work. REFERENCES Olenev, Е.А. (2007). “No conventional use of solid fuel on a railway transportation”, High technologies, 1, pp. 61-66. Olenev, E.A. (2008). “Way of burning and dry distillation of fuel”. Patent 2319065, Russian Federation.
1713
Method of Analytical Calculation Radiating (Steam) Superheater of the Steam Locomotive Е.А. Olenev Vladimir state university, Vladimir, Russia Abstract: The analytical method of calculation radiating superheater is offered, allowing defining temperature of overheated steam in the course of steam locomotive movement. This method can be used for calculation and other heat exchangers.
1. INTRODUCTION Application on a railway transportation of the steam locomotives using cheap fuel – Coal, will allow to lower expenses for transportation of cargoes and passengers. Thus the steam locomotive, being more economic in comparison with diesel locomotives and electric locomotives which work on secondary energy carriers, should be free now from the basic lacks – low EFFICIENCY and heavy conditions of operation. One of ways of increase of EFFICIENCY of steam power plant, as is known, is application overheat steam. Applied on old steam locomotives a convection superheater could not provide a heat of an overheat of steam as they were shielded by a cold surface fire tube through which heat was transferred to boiler water. To raise temperature of an overheat of steam in a steam locomotive it is possible by means of radiating superheater in which heat from combustion products is transferred by radiation and convection. Thus as fuel it is better to use the gas energy received by thermal decomposition of firm fuel directly in a steam locomotive fire chamber (Olenev 2007, 2008). Methods of calculation radiating superheater, resulted in the literature are grafo-analytical, containing subjective estimation elements at a choice and appointment of some settlement parameters that does not allow constructing the mathematical model, giving the chance to spend necessary calculations in real time in the course of automatic control. For the purpose of elimination of the specified lacks, the method has been developed and described more low, which can be applied to calculation and modeling of work radiating superheater and other heat exchangers. 2. DEFINITION OF COEFFICIENT OF A HEAT TRANSFER OF SUPERHEATER
The coefficient of heat emission can be found by radiation under the known formula
αr =
4 ε g C ⎡⎛ Tg ⎞ ⎛ Tw ⎞ 4 ⎤ ⎢⎜ ⎟ −⎜ ⎟ ⎥=
Tg − Tw ⎢⎜⎝ 100 ⎟⎠ ⎣
⎝ 100 ⎠ ⎥ ⎦
2 2⎤ ⎡ ⎛ Tg T ⎞ ⎛ Tg ⎞ ⎛ Tw ⎞ ⎥ ⎟ +⎜ = 10 −2 ε g C ⎜⎜ + w ⎟⎟ ⎢⎜⎜ ⎟ ⎟ ⎝ 100 100 ⎠ ⎢⎣⎝ 100 ⎠ ⎝ 100 ⎠ ⎥⎦
,
(1)
where α r – coefficient of heat emission by radiation, V/(m2·°С); coefficient;
С = σ ⋅108 = 5,67 W/(m2·°K4) – emission σ – a radiation constant, equal
5,67 ⋅ 10 −8 W/(m 2 ⋅ K 4 ) ; ε g – degree of blackness of gas; Т g , Т w – accordingly absolute temperature of radiated gas
and a heat exchanger wall (an element of superheater), °K. Through unit of a surface of the heat exchanger, having on the one hand temperature of smoke gases t g , and with another – temperature of the heated up environment t s (°С), the following thermal stream will be transferred
(
)
q = K t g − ts ,
(2)
where q – heat stream through surface unit, W/m2; K – heat transfer coefficient, W / (m2 °С); t s – temperature of the heated up environment. Same most it is possible to receive, using coefficients heat emission on both parties heat exchanging surfaces
Let us present radiating superheater in a kind of heatinsulated pipes, in which (steam) super heater elements are placed, washed by burning products, moving a countercurrent. Heat exchange process between a wall (steam) super heater element and smoke gases washing it, grows out of joint action of radiation, convection and heat conductivity. The quantitative characteristic of the specified process is the coefficient of heat emission α g = α r + α с ,
978-3-902661-43-2/09/$20.00 © 2009 IFAC
where α r considers influence of emission, and α с – convection and heat conductivity.
′ , q = α g t g − tw
(
)
(3)
′′ − ts ) , q = α s (t w
(4)
′ , t w′′ – accordingly temperature of a wall of the heat where t w exchanger from smoke gases and the heated up environment, °С; α s – coefficient of heat emission on the party of the heated up environment, W / (m2 °С).
1708
10.3182/20090603-3-RU-2001.0507
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
As in metal heat exchangers temperature drop in a wall makes very small size (Several degrees), the thermal resistance of a wall, can be neglected, and ′ = t w′′ = t w . Then, equating the right parts of the consider t w equations (3), (4), we will receive
Summarize the equations (8) and (9), we will receive the expression describing degree of blackness of gas, containing simultaneously carbonic acid and water steam
That can be neglected thermal resistance of a wall tw =
α g t g + α sts . αg +αs
(5)
For calculation the coefficient of heat emission radiation, we will accept more simple dependence of temperature of a wall on temperatures of smoke gases and environment tw =
t g − ts . ln t g t s
(
)
(6)
Taking into account expression (6) and some transformations the equation (1) will become ⎛ ⎞ t g − ts α r =10 −8 ε g C ⎜ Tg + + 273 ⎟ × ⎟ ⎜ ln t g t s ⎝ ⎠
(
)
2 ⎡ ⎛ ⎞ ⎤ 2 ⎜ t g − ts ⎢ ⎟ × Tg + + 273 ⎥ ⎜ ln t g t s ⎟ ⎥ ⎢ ⎝ ⎠ ⎥⎦ ⎢⎣
(
.
(7)
)
As is known, considerable radiating and absorption abilities, having practical value, multinuclear gases possess, in particular, СО2 and Н2О. Unlike solids, the radiation and energy absorption in gases occurs in volume, and their radiating and absorption abilities depend on gas temperature t g , lengths l ways of a beam and partial pressure р. For values pl from 0,006 to 1atm and temperatures from 700 to 2000°С by the author had been received formulas, defining relative radiating ability (or blackness degree) carbonic gas and water steam: ⎡5 ⋅10 −4 ⋅ t g − 3,1 +
ε СО2 = 10
−2 ⎢
ε H 2O = 10
−2 ⎢
(
⎢+ 32,987 − 9,82 ⋅10 −3 t g ⎣
⎡5 ⋅10 −4 ⋅ t g − 3,1 +
(
⎢+ 48,83 − 1,496 ⋅10 −2 t g ⎣
⎤ ⎥, 0, 2788 ⎥ pCO2 l ⎦
(8)
⎤ ⎥ pH 2Ol 0,4313 ⎥⎦
(9)
)( )(
)
)
Where ε СО 2 , ε H 2О – accordingly degree of blackness of carbonic acid and water steam; pCO2 , pH 2O – accordingly partial pressure of carbonic gas and water steam, atm; l – length of a way of a beam, m;
(
)(
)
0, 2788 ⎤ ⎡t − 6200 + 32987 − 9,82t p + g g CO 2 l ⎥. 0, 4313 ⎢+ 48830 − 14,96t p ⎥ g H 2O l ⎣ ⎦
ε g = 10 −5 ⎢
(
)(
)
(10)
Total radiation of such mix is a little bit less than sum of radiations of carbonic acid and water steam, containing in it, as strips of radiation and absorption for СО2 and Н2О partially coincide, leading to mutual partial absorption of energy radiated by them. However at usual parities of components, taking place in smoke gases, the amendment on radiation decrease, possible not to consider. The equation for coefficient of heat emission by radiation will take on form ⎛ ⎜ ⎝
α r = 10 −13 C ⎜ Tg +
⎞ + 273 ⎟ × ⎟ ts ⎠
t g − ts
(
ln t g
)
2 ⎡ ⎛ ⎞ ⎤ 2 ⎜ t g − ts ⎢ × Tg + + 273 ⎟ ⎥ × ⎜ ln t g t s ⎟ ⎥ ⎢ ⎝ ⎠ ⎦⎥ ⎣⎢
(
)
(
)(
⎡t − 6200 + 32987 − 9,82t p g g CO 2 l ×⎢ 0, 4313 ⎢+ 48830 − 14,96t p g H 2O l ⎣
(
)(
)
)0,2788 +⎤⎥ ⎥ ⎦
.
(11)
The coefficient of heat emission by contact is defined basically by conditions of movement and physical properties of gas. At the established turbulent movement of a stream the coefficient of heat emission by contact can be calculated on both parties of a heat-transmitting surface under the known formula
α с = 1,163B
(ωρ )0,8 d 0, 2
= 1,26 ⋅ 10 −3 В
(Vρ )0,8 S 0,2 , A
(12)
where α с – coefficient of heat emission by contact, W/(m2·°С); В – correction multiplier; ω – speed of moving gas, m/s; V – quantity of moving gas, nm3/h; ρ – gas density, kg/nm3, d – the resulted diameter of the channel on which gas moves, m; S – perimeter of the channel on which there is a gas, m; A – live section of the channel, m2. Value of multiplier B calculated from a following parity В = 0,313
λ0,6с р 0,4 η 0, 4
,
(13)
where λ – heat conductivity of gas, W / (m °C); с р – a gas thermal capacity at constant pressure, kJ / (kg °C); η – dynamic viscosity, Pa·s. On the basis of expressions (12), (13) we can write down
1709
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
α с = 3,95 ⋅10 −4
λ0,6с р 0,4 (Vρ )0,8 0,2 S . A η 0, 4
η а = 15,5 ⋅ 10 −6 + 0,28 ⋅ 10 −6 t a 0,69 .
(14)
For smoke gases at humidity 5-20 % in the range of temperatures from 0 to 1500°С have been received following dependences of coefficients of a thermal capacity c pg , kJ / (kg °C), heat conductivity λ g , W / (m °C), and viscosity
η g , Pa·s, gases from their temperature. c pg = 1,032 + 3,13 ⋅10 −4 t g 0,865 ;
λ g = 2,21⋅10 −2 + 4,65 ⋅10 −5 ⋅ W +
(
)
+ 7,56 ⋅10 −5 + 6,98 ⋅10 −7 ⋅ W t g
;
α сu =
(17)
where W – humidity content of gas, %. At pressure of water steam from 12 to 20 kg/sm2, characteristic for locomotive boilers, in the range of temperatures 240 ÷ 500 °C thermal capacity c pu the pair can be described the formula с рu =
16,34е
0,094 рu
+ 1,957tu + 66,2 , tu
(18)
kg/sm ; tu – steam temperature, °С. Heat conductivity λ u and dynamic viscosity ηu of steam in the specified range of pressure and temperatures from 200 to 700 °С will accordingly be expressed
ηu = 8 ⋅ 10 −7
(tu + 273)1,632 .
c pa = 0,9897 + 10 t a + 3,243 ⋅ 5,8
.
(22)
Heat conductivity of air at temperature from 0 to 1500 °С
λа = 2,44 ⋅10
−2
−4
+ 1,45 ⋅10 ta
0,865
,
0, 4
(tu + 273)0,5832 . (tu + 946)0,2
(Vρ )0,8 S 0,2 (168,3+t A
a
)
0,865 0,6
⎛ 0,9897 + 10 − 4 t + 3,243 ⋅5,8 −(3+0,004ta ) ⎞ a ⎟ ×⎜ −6 −6 0,69 ⎟ ⎜ t 15 , 5 10 0 , 28 10 ⋅ + ⋅ a ⎠ ⎝
(26)
×
(27)
0, 4
.
)
(28)
where qs – quantity of heat received by the environment, kJ/h; Vs – quantity of environment, the past through the heat exchanger, m3/h; c eps , t se and c bps t sb – accordingly a thermal capacity, kJ / (kg °C), and temperature, °С, environments on an exit and input of a heat exchanger; ρ s – density of environment, kg/m3.
and
dynamic viscosity η a of air at atmospheric pressure in the range of temperatures from 0 to 1500 °С can be expressed as follows − (3+ 0,004ta )
3,122⋅105 A
α сa =1,965⋅10 −6
qg =
The thermal capacity c pa , heat conductivity λa
−4
(Vρ )0,8 S 0,2 ×
1,665 ⎤ ⎡ 5 +⎥ ⎢4,578⋅10 + 550 −tu ×⎢ ⎥ ⎢ + 65⎛⎜ pu (1000 −tu )+tu ⎞⎟ ⎥ ⎢⎣ ⎠ ⎥⎦ ⎝ 10
(21)
tu + 946
)
If through γ to designate a share of heat lost through walls radiating of superheater, the quantity of heat which smoke gases should give, will turn out equal
(20)
tu + 946
0, 4 (25) ⎛ 3297 +t 0,865 ⎞ g ⎟ ⎜ 61,25+t 0,724 ⎟ g ⎝ ⎠
0, 6 ⎜
qs = Vs ρ s c eps t se − c bps t sb ,
2
(tu + 273)2,06 ,
A
(
(19)
where ðu – pressure of steam in superheater elements,
λu = 1,088 ⋅10 −4
(Vρ )0,8 S 0,2 ×
Let's establish now interrelation between target temperatures of smoke gases and the heated up environment. The quantity of heat received by the environment in the heat exchanger will be equal
and in a range from 400 to 700 °С ⎡ 1,665 ⎤ +⎥ 4,578 ⋅105 + 550 − tu , c pu = 4,19 ⋅10 −6 ⎢ ⎢+ 65(0,1 p (1000 − t ) + t ) ⎥ u u u ⎣ ⎦
α сg = 1,408⋅10 −6
(
(16)
η g = 14,7 ⋅10 −6 + 0,24 ⋅10 −6 t g 0,724 ,
On the basis of the resulted expressions we will receive following equations for coefficients of heat emission by contact for smoke gases α cg , steam α cu and air α ca
× 31661,9 + 66,62W + (108,3 + W )t g
(15)
(24)
(
Vs ρ s c eps t x − c bps t sb 1− γ
) = V ρ (c g g
b b pg t g
− c epg t ge
)
(29)
where q g – the quantity of heat given by smoke gases, kJ / h; Vg – quantity of smoke gases, the past through the heat
exchanger, m3/h; c bpg , t gb and c epg , t ge ; – accordingly a thermal capacity, kJ / (kg °C), and temperature, °С, smoke gases on an input and exit of a heat exchanger; ρ g – density of gases, kg/m3.
(23)
1710
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
In the range of temperatures from 100 to 2000 °С for average structure of smoke gases, it is possible to express dependence of their thermal capacity on temperature
αs =
c pg = 1,038 + 1,12 ⋅10 −4 t g .
Hence, the coefficient of a heat transfer will become
(30)
On the basis of expressions (29) and (30) we will write down
(
)
2⎞ ⎛ 2 1,12 ⋅10 −4 ⎜ t ge − t gb ⎟ + 1,038 t ge − t gb + ⎝ ⎠
+
(
Vs ρ s c eps t se − c bps t sb Vg ρ g (1 − γ )
)=0
ts =
Vs ρ s
(1-γ )Vg ρ g
(c
e e ps t s
)
2
− c bps t sb + 1,12 ⋅10 − 4 t gb + 1,038t gb −
− 4634.
In a radicand, in subtracted, instead of a thermal capacity
c eps it is necessary to substitute function of its dependence on temperature, i.e. for steam it is the equations (18) or (19), and in case of heating, for example air – (22). After substitution we receive value of temperature of smoke
t se
t se
environment, i.e. being set by value , on the equation (32) we calculate the corresponding temperature of smoke gases on an exit. Having established, thus, interrelation between temperatures on an input and exit of a heat exchanger, using the equations (25), (26), (27), it is possible to define values α c for the concrete environment and α r from expression (11). After that for smoke gases the coefficients of heat emission α g is calculated. To true values of temperatures t se and
t ge
will correspond such coefficients of heat emission on
both party of a heat-transmitting surface which being are substituted in expression of coefficient of a heat transfer К, will provide equality
(
3,6 ⋅ К ⋅ Аw ⋅ t s − Vs ρ s c eps t se
− c bps t sb
)= 0 ,
(33)
where t s – average value of temperature of environment in the heat exchanger, °С; Аw – the area of a surface of heat exchange, m2. Let’s accept change of coefficients of heat emission along the heat exchanger under the logarithmic law then the average coefficient of heat emission is equal
αg =
α gb
− α ge ln α gb / α ge
(
)
,
(34)
e s
− α sb
(35)
(α − α )(α − α ) )ln(α / α )+ (α − α )ln(α b g b g
e g
e s
e g
b s
b g
e g
t se − t sb
(
ln t se / t sb
e s
/ α sb
),
).
(36)
(37)
(
)(
3,6 ⋅ Аw α gb − α ge α se − α sb
α se
(32)
gases on an exit from the heat exchanger depending on
(α
)
Substituting expressions (36), (37) in the equation (33), we will receive
(
= 94,5 ×
× 2405 −
(
We accept change of temperature of environment in the heat exchanger under the logarithmic law. Then
(31)
The reference temperature of smoke gases and environment, as a rule, is known, therefore, solving the equation (31) concerning temperature of smoke gases on a heat exchanger exit, we will have t ge
К=
α se − α sb . ln α se / α sb
−α sb
)(
ln α gb /α ge
)(
+ α gb −α ge
)
)ln(α
(
e b s /αs
)×
)
te − tb × s e sb − Vs ρ s c eps t se −c bps t sb = 0 . ln t s / t s
( )
(38)
Using dependences (15), (18), (19) and (22) thermal capacities of the heated up environment from temperature, having substituted this dependence instead of a parity
(c
e e ps t s
)
− c bps t sb and being set by value t se from the equation
(32) find t ge , then calculate on the equations (25), (26), (27), (11) values α s , α r and then α g to begin with and end of the heat exchanger. Then check justice of expression (38) which allows to judge true value of temperature t se . The equation (38) gives the chance to receive also value of temperature replace all
t se directly. For this purpose it is necessary to variables with their functions. For example,
instead of α gb to substitute expression from the equations (11), (25) ⎞ ⎛ t gb − t se ⎟× 10−13 C ⎜ t gb + 273+ + 273 b e ⎟ ⎜ ln t g t s ⎠ ⎝ 2⎤ ⎡ b e ⎞ 2 ⎛ t g − ts ⎢ ⎟ ⎥× × ⎢ t gb + 273 + ⎜ + 273 ⎜ ln t b t e ⎟ ⎥ g s ⎝ ⎠ ⎥⎦ ⎢⎣
(
(
)
(
(
)
)
)(
⎡t b − 6200 + 32987 − 9,82t b p g g CO 2 l ×⎢ 0, 4313 ⎢+ 48830 − 14,96t b p H 2O l g ⎣
(
+ 1,408⋅10−6
1711
)(
)
)0,2788 +⎤⎥ ⎥ ⎦
(Vρ )0,8 S 0,2 ⎛⎜ 31661,9 + 66,62W + ⎞⎟ ⎜ + (108,3 + W )t gb ⎟ A ⎝ ⎠
+ 0,6
×
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
⎛ b 0,865 ⎞ ⎜ 3297 + t g ⎟ ×⎜ ⎟ 0 , 724 ⎜ 61,25 + t gb ⎟ ⎝ ⎠
equation (36), W / (m2 °C). The temperature of a heattransmitting wall according to expression (5) in the beginning
0, 4
.
(39)
superheater will make
t wb = 843,7 °С, in the end –
and t s on etc. Thus Having replaced thus t g on bulky enough equation having, however, only one unknown
t we = 511,4 °С. The average temperature of steam in superheater according to the equation (37) will be equal t s = 530,3 °С
size tse which dares a dichotomy method will turn out. Such way is simpler and is comprehensible for the COMPUTER.
4. AN EXAMPLE 2. DEFINITION OF TEMPERATURE OF HEATING OF AIR
Having received value of temperature t se , then define all other necessary parameters.
To define temperature of air leaving the heat exchanger
tse ,
t gb
3. AN EXAMPLE 1. DEFINITION OF PARAMETERS OF THE OVERHEAT STEAM To define temperature leaving of superheater steam having reference temperature tub = 437 °С and pressure of pu = 16 kg/sm2. The expense of steam of Vu ρu = 4000 kg/h, quantity passing through superheater smoke gases Vg = 2000 m3/h. The share of heat lost through walls radiating superheater, makes γ = 5 %. Smoke gases contain 19 % СО2, 1 % Н2О, and 5 % of moisture and are included into the heat exchanger with temperature t gb = 1400 °С. Superheater is executed in the form of the cylinder in diameter of 1,1m in which are placed 80 tubes in the size of 24/30 mm for steam passage, having perimeter of Su = 6,03 m and live section Au = 0,0362 m2. Channel perimeter, on which движутся smoke gases, S g = 11 m, and its live section Ag = 0,8938 m2. Surface of heating Аw = 18,9 m2.
The decision of the equation (38) by method of a dichotomy, gives size of temperature of the leaving steam from the superheater tuе = 636,1 °С. Substituting the received value in the equation (32), we find temperature of smoke gases on an exit t ge = 874 °С. By means of expressions (25) and (26) we calculate coefficients of heat emission by contact to begin with and end the heat exchanger (For smoke gases and steam) = 6,16
е α cu
2
W/(m ·°С),
2
= 230,46 W/(m ·°С),
b α cu = 206,26 W/(m2·°С). Under the formula (11) it is had
α rb α gb
= 79,85
2
by smoke gases with reference temperature t gb = 1050 °С, containing 19% СО2, 1 % Н2О and 5 % of a moisture. Quantity of warmed up air Vа = 2500 m3/h. Volume of the smoke gases which are passing through the heat exchanger Vg = 2900 m3/h. Smoke gases moving in the cylinder in diameter of 1,1m, having perimeter S g = 3,456 m, and live section Ag = 0,95 m2. Outside of the cylinder the ring channel in width of 19 mm for air passage is placed. In the air channel are established 24 directing edges, by 14 × 6 mm section. Considering equal 6 mm a thickness of metal of the cylinder, we receive the internal diameter of the ring channel equal of 1112 mm and external diameter, equal 1150 mm. From here perimeter of the air channel of S а = 7,78 m and live section Aа = 0,0655 m2. The share of heat lost through walls of the heat exchanger, makes γ = 15 %. The area of a heat-transmitting surface Аw = 12,9 m2. The effective thickness of a radiating layer, product pCO2 l , pH 2Ol will be same, as well as in the previous example.
The effective thickness of a radiating layer for the cylinder in diameter of 1,1m will be equal to l = 0,9 ⋅1,1 = 1 m, therefore pCO 2 l = 0,19 atm, pH 2Ol = 0,01 atm.
b α cg
having reference temperature tаb = 420 °С. Air is warmed up
W/(m ·°С),
α re 2
2
= 37,03 W/(m ·°С).
= 79,85 + 6,16 = 86,01 W/(m ·°С),
α ge
Then
= 37,03 + 5,33
= 42,36 W/(m2·°С). According to expressions (34), (35) average coefficients of heat emission for smoke gases will be equal α g = 61,63 W/(m2·°С), for overheat steam W / (m2 °C). The heat transfer coefficient is defined from the
Solving the equation (38) we find temperature leaving the heat exchanger of air tае = 608,77 °С. Substituting this value in the equation (32), we find t ge = 889,8 °С. By means of expressions (25) and (27) we calculate coefficients of heat emission by contact to begin with and end b e = 5,65 W / (m2 °C), α cg = 5,38 W / heat exchanger α cg b e (m2 °C); α ca = 61,32 W / (m2 °C), α ca = 64,93 W / (м2 °C).
Under the formula (11) it is had α rb = 54,47 W / (m2 °C),
α re = 37,43 W / (m2 · °C). Then α gb = 54,47 + 5,65 = 60,12 W / (m2 °C),
α ge = 37,43 + 5,38 = 42,81 W / (m2 °C). According to expressions (34), (35) average coefficients of heat emission for smoke gases will be equal α g = 50,98 W / (m2 °C), for air α a = 63,1 W / (m2 °C). According to expression (36), K = 28,12 W / (m2 °C). The temperature of
1712
13th IFAC INCOM (INCOM'09) Moscow, Russia, June 3-5, 2009
a heat-transmitting wall in the heat exchanger beginning will b make t w = 820,9 °С, in the end – t we = 613,2 °С. From the equation (37) average temperature of air will be equal t s = 508,6 °С.
5. CONCLUSION The stated method includes performance of sequence of formal operations, owing to what completely all calculation of the heat exchanger can be carried out on the computer. The received mathematical model, gives the chance to spend in real time necessary calculations for automatic control of boiler work. REFERENCES Olenev, Е.А. (2007). “No conventional use of solid fuel on a railway transportation”, High technologies, 1, pp. 61-66. Olenev, E.A. (2008). “Way of burning and dry distillation of fuel”. Patent 2319065, Russian Federation.
1713