Exergetic efficiency substances
of comminution
of solid
Ryszard Petela lnstytut Techniki Cieplnej Politechniki Slaskiej, Gliwice, Poland (Received 17 December 7982) Based on literature data, a short description of a method for determining the internal energy increase of material due to enlarging its surface area in a comminution process, is given. An exergy balance of the comminution of a solid substance, has been discussed. A method for determining the exergetic effect of comminution, has been presented. Generalized exergetic efficiency of comminuting installation has been defined and the efficiency of the common cases of comminution has been discussed. Examples of exergetic efficiency values of pneumatic jet mill comminuting sand and of EM-70 mill for coal, have been calculated. (Keywords: comminution; solid substances; exergetic efficiency)
Comminution of solid substances is a widely used process, for example, the grinding of coal. The aim is to enlarge the surface of a given amount of substance which in the case of a fuel increases its burning efficiency. One measure of the effect of comminution is the increase in internal energy of the cornminuted material, which is due to an increase in only one component of the internal energy. This component is the potential energy of mutual interaction between particles. The dependence of the energy, e, of potential interaction between two particles on the distance, 1, between them is shown in Figure 1. With the growth of 1from zero to infinity the energy, e, decreases to the minimum negative value, e,, , in the equilibrium state (for I,) and then increases asymptotically to zero. The negative quantity E’is the bond energy of one particle in the state of equilibrium in relation to another particle. The increase in internal energy due to the increase in surface area during comminution can be determined from Mielczarek’s theory’. The location of particles is presented schematically in Figure 2, taking as an example a body in the form of an hexahedron located in vacuum. The bond energy of particle, A, inside the body amounts
to E. The bond energy of particle, B, on the surface amounts to s/2 because the plane of this surface divides the space around the particle into two half-spaces, one of which is empty. The edge is made by intersection of two planes so the bond energy of particle C on the edge amounts to a/4. The vertex is arrived at by successive bisection so that the bond energy of particle D on the vertex amounts to s/8. The bond energy, E, of all the n surface particles: n=n,+n,+n,
(1) where: n,, n,, n, = the number of particles on the surface, edges and vertices, respectively, is determined as follows:
If on one edge of an hexahedron there appears to be x
1 I
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0016-2361/84/03041W-05$3.00 @ 1984 Butterworth& Co. (Publishers)
414
Attractlon
Ltd
FUEL, 1984, Vol 63, March
,’
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I’ 1’ /
Figure 2 Possiblelocation of particles:A, inside; B, surface;C, edge; D. vertex
Exergetic efficiency of comminution of solid substances: R. Petela
particles then, from Equation (1) one receives n=6(x-2)2+12(x-2)+8
(3)
and from Equation (2): E=;[6(n-2)2+6(x-2)+2]
(4)
The greater is the number x the better are the approximations: n zn, and E xn,&/2 or: E=nq
(5)
The ratio of nAof number n of surface particles to the area A of this surface n
nA=A
is for given substance practically constant (nAxconst) because it does not depend significantly on temperature. Substituting Equation (6) into Equation (5) yields: E=An,t
(7)
The concept 0 determined as follows: a=n,?
(8)
2
has dimension, J me2 , and presents the surface tension on the phase boundary. This tension on the solid substance surface, for example, in a case when the substance is surrounded by a gas, is different from a case where the substance is surrounded by a liquid which is generally more dense than a gas. The quantity 0 is a function of temperature. After taking into account Equation (8) Equation (7) takes the form: E=aA
(9)
Consider now the two states of substance, i.e., before comminution (subscript 1) and after comminution (subscript 2). The stable internal energy increase AUAresulting from the growth of surface area of the comminuted solid substance is equal to the increase AE= E2 -E, of the bond energy of surface particles (AU, =AE). Using Equation (9) the following equation can be derived: AU,=a2A2-alAl
(10)
Comminution of a substance is obtained when a certain energy, EC, is delivered. In practice, this energy is readily supplied in the form of mechanical energy, e.g. in form of kinetic energy or by means of the work of crushing. However, the effect of the stable increase in internal energy (AU,>O) of the comminuted substance can be obtained when the energy EC, used to comminute, is larger than a certain barrier value Eb, (EC> E,,), which, for example, in a case of kinetic energy acting on the material with the area A, is determined as follows: V-taA
&,=&-
elastic strain. The expression aA is the energy of the broken bonds in the particles on the surface of the comminuted substance and because of this energy these particles become detached. When the supplied kinetic energy is too small (EC< Eb) the solid substance is not comminuted and the energy E, is only used to preheat the substance and can be lost to the environment. EXERGY BALANCE OF COMMINUTION PROCESS Solid substance comminution is a process of energy conversion. To carry out an analysis of the efficiency of such a process it is best to apply an exergetic analysis’ -4, which has advantages over an energetic analysis. For example, by using exergy the real value of compressed air at ambient temperature can be estimated more accurately than by using energy (enthalpy). An exergetic analysis is based on the exergy balance of the system under consideration, which is the arbitrary cornminuting installation (Figure 3) in steady state. Therefore, there is no system exergy increase. To the installation is delivered the exergy B, of the solid substance which is to be cornminuted. Other possible delivered exergy includes the exergy B,, of the gases which can be the driving agents (e.g. compressed air, water vapour) of the process; and the mechanical or electrical exergy B,. If the installation is in any way preheated then the exergy drop, -B,,, of the external heat source should be taken into account. Comminution of a solid substance can be considered as a physical process and the calculations of exergy take into account only those components which change in a physical process. Therefore, the exergy, B,, of delivered material consists of the physical exergy, B,,; kinetic exergy, B,,; potential exergy, B,,; and of the separated exergy, BIA, of the bonds of surface particles: B, =B,r+B,,
(12)
+B,, +BIA
After comminution, the material leaving the installation has an exergy correlating to Equation (12) as follows: B2
=
B2r
+ B2k
+
82,
(13)
+ B2,
The exergy of each driving agent is calculated with respect to Equation (12) but without the component concerning the exergy of the surface particles bonds B,, = Ba,f + Balk+&I,
(14)
42
(15)
=
Ba2f
+
Baa
+
Ba2p
Commwwting installotlon
(11)
Y
where: y= the compressive stress on the failure barrier; EY =Young’s modulus; and V= the volume of comminuted material. The expression yV/2E, is the energy of maximum
/exergy
external loss
Figure 3 Schematic diagram of the exergy balance of an installation for cornminuting of solid substance
FUEL, 1984, Vol 63, March
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Exergetic efficiency of comminution of solid substances: R. Petela
The physical exergy, Br, of each substance results from the difference of temperature, 7: and pressure, p, of this substance in relation to the respective values of T, and pe of this substance in thermodynamic equilibrium with the environment. The exergy, Br, is calculated from the equation : Bf=Z-Z,-T,(S-Se)
(16)
where: I, I,, S, S, = enthalpy and entropy of substance in considered state and in the state of thermodynamic equilibrium with the environment, respectively. Kinetic exergy, B,, of substance equals its kinetic energy Ek of translatory and rotary motion with the velocity determined in relation to environment Bk=Ek
B,=E,
(18)
If, for a given rate of cornminuted material, the product 02A2 is the maximum barrier value, independent of the value a,A, (i.e., on the surface area of delivered material), then the efficiency, q, increases as the product o,A, Al =O, so:
AB, = BzA - BIA
(19)
is the useful effect of comminution stable internal energy increase
and is equal to the
AB,=AU,
(20)
The rate of mechanical or electrical exergy, &,,, is equal to the mechanical or electrical power output N: B,=N
(21)
It is possible that the cornminuting installation is not preheated but cooled and then one should take into account the heat loss expressed by means of the exergy increase, B,,, of external heat source. The quantity B, is calculated from : T-T (22)
where: Q = heat absorbed (negative) or lost (positive) by external heat source; and T= temperature of this heat source. In the exceptional case when no heat is exchanged (absorbed or lost) between the installation and the natural environment (T= T,), Equation (22) gives B, =O. The exergy external loss, 6B,,, which can happen to the installation consists of the following quantities: 6B,, = & + Bzr+ & + Bzp+ B,,
(23)
As there is no law of exergy conservation, the exergy balance must be closed up by the exergy internal loss 6B, due to the irreversibility of the process under consideration, 6B is calculated from the equation cSB=TJI
(24)
where: II =sum of the entropy increases of all the substances and heat sources taking part in the process. The exergy balance equation for a cornminuting installation therefore takes the form: =B, +B,, +B,, +SB
FUEL, 1984, Vol 63, March
fJ24
rlm”x=B, +B,, +B,-B,,
-B,,
(27)
This means that the successive grinding of the whole of the material in the same installation will not reduce the grain size, but it is useful to return the separated large grains back to the same installation. One example of this cornminuting installation is a pneumatic jet mill driven by compressed air. In practice, one can assume a constant grain temperature (gl = cr2= 0) equal to the environment temperature. Neglecting kinetic and potential exergy Equation (26) yields:
a(ii,-4)
‘=
i;,b,,
(28)
where: k,, k, =material surface area delivery rate before and after grinding, respectively; G, = air flow rate; and b,, = specific exergy of air. The quantity b,, is calculated from Equation (16) assuming that the air is an ideal gas:
b,, =c,K, - r,) -T,
c,ln$-R,lnP”’ e Pe ) ( where: cP= the specific heat of air at constant pressure; T,, = air temperature; R, = individual gas constant for air; and pal, pe = the absolute pressure of air and the environment, respectively. Another example of a cornminuting installation is a mill (e.g. EM-70) where the air introduced and the electric driving power N should be considered. In such a case it is assumed that the grain sizes in the mill are very much reduced (A, > A,) and the rate A, can be neglected (A, = 0). In accordance with Equation (27) one obtains the value q= qmaxcalculated as follows: (30)
(25)
This equation is the basis of the exergetic analysis of the comminution process and the basis for definition of the exergetic efficiency.
416
(26)
decreases. The maximum value, q,,,, is reached when
The exergy increase
B, +B,, +B,-B,,
OF COMMINUTION
Equation (25) is presented schematically in Figure 3. Using the appropriate interpretation for the terms in this equation one can define the exergetic efficiency, 4, of the cornminuting installation. The equation for q should contain all the items except the losses. Taking into account the aim of comminution, the efficiency numerator should contain only the increase, AB,, in the bond energy of the surface particles, calculated from Equations (20) and (10). All the other items from Equation (25) are the components of the driving exergy and should be put into the efficiency denominator:
(17)
The potential exergy, B,, of substances is equal to its potential energy, E,, calculated in relation to the environment level height:
B,=Q’
EXERGETIC EFFICIENCY INSTALLATION
It should also be noted that when the liauid is comminuted (atomized) the equations are applied with the assumption that A, =0 because liquid, before atomization, is homogeneous.
Exergetic efficiency
VALUES OF EFFICIENCY In the literature there are relatively few experimental data which are complete enough to allow a calculation of the exergetic efficiency of the comminution of a solid substance. Complete data sets are often obtained with large errors of measurement, particularly those pertinent to the area rates A, and A,. Also large scatter results appear in the values of 0 determined using the various experimental and analytical methods. Therefore the absolute values of efficiency, I], are usually calculated with a large margin of error. In spite of this one can state that the values rl are relatively small and this proves that the solidaubstance comminution process is irreversible. In spite of the unreliable values, q, the influence of various factors on the process can be estimated quite well on the basis of the relative increases in values of 17. The measurements have been carried out’ for the pneumatic jet mill. The compressed air has been delivered at the pressure pa, =0.4 MPa and at the temperature T,, equal to the environment temperature T,, = T, = 286 K. The cornminuted material.was sand (SiO,). The value, 0, for sand is estimated’ to be within the range 2-228 J rne2. The large values are obtained by measurement and the small ones by calculation. The calculated value is more trustworthy6, so it was assumed that u = 2 J rnm2. The air flow rate G, = 394 kg h - ’ was constant. The calculations of q were carried out using Equations (28) and (29) and the results obtained are presented in Figure 4, against the serial number i of successive millings of the same charge. For comparison the diagram also shows the value of maximum efficiency, qmax,calculated from Equation (28) with the assumption that A, -0 and for the value A, obtained after milling (i = 1). The increase (A2i- Ali) of the grain surface area rate for the ith milling is_calculated by substitution, Ali = Azi _ 1. The increase in A2 after the second and all subsequent millings (i > 1) is small and this confirms the existence of a barrier mentioned earlier. The consequence of this is an efficiency value near zero. During other experimental investigations7 on the comminution of sand (a = 2 J me2) the pneumatic jet mill was fed each time with material of variable initial grain coarseness and at variable values of the air flow rate, G,, and air pressure, pa,. The temperature of this air was
of comminution
of solid substances:
R. Petela
constant (T,, = T, = 286 K). The efficiency values, u], calculated from Equation (28) do not change smoothly (Figure 5) with the growth of pressure pa, because the initial grain size was different each time. For comparison, the maximum efficiency values, q,,,,,, for the same cases, have been calculated from Equation (28) at Al = 0 and are shown in Figure 5. The values, )I,,,~~,diminish monotonically with the growth of compressed air pressure, pa,. The efficiency of comminution of coal has been calculated from Equation (30) using measured results8 for the mill EM-70. The mill was fed with coarse grains (A, $ A,) so that A, = 0 is assumed in calculations. Therefore, the calculated efficiency value, q, is the maximum efficiency (q = q,,,). The value cr= 11.4 J m - * is assumed on the base of literature data’. The specific exergy b,, of air is calculated from Equation (29) for the following data: pa, =0.108 MPa; T,, =623 K; and T,= 286 K. The moisture content of the delivered coal was 11 wt%, and 4 wt% in the resulting grains. However, to simplify the problem the drying process of coal was neglected. The calculated results are presented in Figure 6 on the characteristic diagram for the mill. The influence of particular factors on the exergetic efficiency q of the mill is shown. For example, at a constant effect of comminution
V.7
a=
2 Jm-’
To, T, = 286K q
--o--
0.3
04
4rrw -q
0.5
-__
06
0
P,,, (MPa)
Figure 5 Influence of driving compressed air pressure p,, on the exergetic efficiency, tf, and qmaxof sand comminution CJ=Z
JmeZ
i;, = 394 kg h-’ PO,/Pe
q
4
T,, = T, = 286 K
I
30
./ 0
2
4
6
8
10
Figure 4 Exergy efficiency, 1, of sand comminution depending on the serial number, i, of successive grinding
$
1
32
34
I
I
36
38
I
40 R 88 %
I
I
I
I
42
44
46
48
I
50
Figure 6 Characteristic diagram.for EM-70 mill showing the values of exergetic efficiency r~. G,, wet coal rate delivered to the mill, t/h; e,, unitary consumption of electrical energy to drive the mill, kWh/t; Ras, mass of grains which do not pass through the sieve with size mesh of 88 pm, %; G,, air flow rate, t/h
FUEL, 1984, Vol 63, March
417
Exergetic efficiency of comminution of solid substances: R. Petela Mill EM-70 r--i
&=C$
c(T,-T,)-T,clng [
1
e
(31)
where: c=specitic heat of coal dust. The air exergy A,, delivered to the mill is a relatively large part (87.7%) of the entire driving exergy. If one assumes that this air does not influence the comminution process then a certain net value, qN, of comminution efficiency determined from formula (32)
a
“8,
Mill EM -70
“T,,
= 1.16%
/I
can be taken into consideration. In the case illustrated in Figure 7a this value amounts to qN= 2.9112.3 = 23.7%. This value is considerably larger than the pneumatic jet mill efficiency. For comparison, Figure 7b presents a schematic diagram of the simplified energetic balance for the same mill and for the same point M. The sum of electric power, N, and enthalpy Z,,_of delivered air is equal to the _sum of the energy rate AU,, enthalpy I,, of air, enthalpy I, of coal dust and the heat Q, lost to environment. The energetic efficiency of the mill is expressed by the equation: AU, ‘en=N+Z,,
I
The net value, v._,~, of energetic efficiency is equal to the respective exergetic efficiency and amounts to qenN= AOJN = qN=23.7x. However, the other items in the energetic and exergetic balances differ considerably.
I
I I
CONCLUSION
b Figure 7 An example of the exergetic balances for the EM-70 mill
(a) and energetic
(b)
(R,, = con$) the greater efficiency, 1, the greater is the mill capacity, G,. The process occurring in the mill is complex because it consists not only of coal comminution but also, e.g., of coal drying, preheating, grain separation, coarse grain circulation, the entrainment of coal dust mixed with air, etc. In such a complex process the efficiency definition of the coal comminution alone is a problem. The problem can be illustrated by the diagram of an exergy balance shown in Figure 7u, which, for simplification, is calculated taking into account only the most important balance items. The diagram is concerned with the mill output determined by the point M in Figure 6. The temperature of air and of the coal dust carried off from the mill are equal (K, = q = 373 K). The air exergy Ba2 is. calculated from Equation (29) and the coal dust exergy B, from the following:
418
(33)
FUEL, 1984, Vol 63, March
This Paper introduces for the first time the concept of exergy into the analysis of comminution process of solid substances and shows the order of magnitude of the process exergetic eficiency. The Paper suggests a means of obtaining a more exact analysis to improve or optimize the construction, design and exploition of any comminuting installations. REFERENCES Mielczarek, E., Habilitation Thesis, Zesz. nauk. Politechniki Czestochowskiej, Czestochowa, 1981 Szargut, J. and Petela, R. Egzergia, WNT, Warsaw, 1965 Szargut, J. and Petela, R. Eksergija, Energija, Moscow, 1968 Szargut, J. Energy 1980, 5, 709 IKiEM Nr 3, Politechnika Rink, R. Prace naukowe Wroclawska, Wrodaw, 1970, pp. 45549 Mielczarek, E. Politechnika Czestochowska, Cmtochowa 1982, personal communication Mielczarek, E. Doctoral Thesis, Politechnika Czestochowska, Czestoehowa, 1973 Poradnik tennoenergetyka, WNT, Warsaw, 1974 Romadin, W. P. ‘Pyleprigotowlenije’, Gozenergoizdat, Moscow, 1953