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Elementary theory of stationary vortex columns for solar chimney power plants
Available online at www.sciencedirect.com
Solar Energy 83 (2009) 462–476 www.elsevier.com/locate/solener
Elementary theory of stationary vortex columns for solar chimney power plants N. Ninic *, S. Nizetic 1 Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Department of Thermodynamics, Thermotechnics and Heat Engines, University of Split, R. Boskovica bb., 21000 Split, Croatia Received 12 December 2007; received in revised form 30 August 2008; accepted 14 September 2008 Available online 9 October 2008 Communicated by: Associate Editor S.A. Sherif
Abstract This paper aims to develop and make use of the availability of warm, humid air. In particular, we focus on the possibility that this availibility can be concentrated at the ground level without using a solid ‘‘chimney”. The results reveal that this concentration can be achieved via the formation of an updraft ‘‘gravitational vortex column” (GVC) situated over turbines. A simplified physical and analytical GVC model is developed in this paper. A numerical solution is given for a characteristic case, with a GVC process as a part of the cycle, similar to the Brayton cycle obtained in a gravitational field. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Solar chimney; Vortex; Availability; Numerical modelling
1. Introduction The concept of a solar chimney power plant (SC) was originally proposed by Schlaich (1968). From 1980 to 1989, a prototype plant in Manzanres was developed, constructed, and tested (Haff et al., 1983; Haff, 1984; Schlaich, 1995). A detailed literature review of this field is given in Bernardes et al. (2003). According to Padki and Sherif (1989) and Schlaich (1995), desert areas and subtropical zones are of primary interest for medium and large SC plants. Solar chimney power plants can also be used under exceptional conditions (Bilgen and Rehault, 2005; Nizetic et al., 2008). The role of a concrete chimney in an SC power plant is in the concentration of a part of the working availability (of heated air leaving a solar collector) at the chimney inlet. *
Corresponding author. Tel.: +385 21305955; fax: +385 21463877. E-mail addresses: [email protected] (N. Ninic), [email protected] (S. Nizetic). 1 Tel.: +385 21305948; fax: +385 21305954. 0038-092X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2008.09.002
Part of the working availability used by the chimney is concentrated at the bottom of the chimney and is in the form of the mechanical non-equilibrium. The pressure in the chimney is lower than the atmospheric pressure at the same level. That form of non-equilibrium can be extracted using turbines. In this paper, the idea for the replacement of a chimney (of a limited height) with a vertical gravitational vortex column is proposed. In the optimal case, the height of the vortex column would be equal to the thickness of the troposphere layer. At the bottom of this stream structureatmospheric gravitational vortex (GVC) (Ninic, 2006), a significantly low pressure (compared to the atmospheric pressure at the collector outlet) would be achieved. The idea of using an atmospheric gravitational vortex as a flow object to play the role of a convective chimney is not a new one. It was originally presented in Dessoliers (1913). In the 1950s, Dessens performed an experiment that, according to Dessens (1962, 1969), proved that such an idea was possible in principle. The idea of using an atmospheric gravitational vortex as a specific heat engine was proposed and somewhat developed by Michaud (1999,
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Nomenclature a ek ep enet coll tech g h h+ M m_ m_ R m_ 01 m_ 12 P p Q_ coll Q_ out R s t m wt wtc wtp wc wfr wR wz 0 1 wz 1 2 wz x zc
angular momentum (m2/s) specific kinetic energy (J/kg) specific potential energy (J/kg) the net technically feasible part of exergy or height potential (J/kg) gravitational acceleration (m2/s) specific enthalpy of air (J/kg) specific total enthalpy (J/kg) momentum (Nm) mass flow rate (kg/s) radial mass flow rate (kg/s) mass flow rate through the internal GVC layer mass flow rate through the GVC updraft shell resultant turbine power (MW) pressure (Pa) heat absorbed in the collector (J) rejected heat (J) radius (m) specific entropy (kJ/kgK) temperature (°C) specific volume of air (m3/kg) specific shaft work (J/kg) specific shaft work in the central turbines (J/kg) specific shaft work in the peripheral turbines (J/ kg) circular velocity (m/s) specific internal friction work (J/kg) radial velocity (m/s) vertical velocity (m/s) vertical velocity of the internal GVC layer vertical velocity of the GVC updraft shell mixing ratio (kgd.a/kgv.) – mass of moisture per unit mass of dry air outlet height (m)
2005). Specifically, Michaud (1975) proposed a ‘‘vortex power station” with heated air, creating a vortex in the atmosphere. The previous works of Haff et al. (1983), Michaud (1995) and Ninic (2006) investigated the magnitude of the working availability of warm air in the atmosphere. In the last study, the term for the technically useful part of the working availability of collector air was worked out in detail. It was defined as the technically feasible part of the working availability in a gravitational field: work gained by reversible adiabatic expansion to a state of stable mechanical equilibrium with the surrounding atmosphere (at a maximum accessible height of zmax). The technically feasible part was defined by
enet coll tech
¼
Z 0
zmax
dpa ðzÞ mcoll ðzÞ g dz: dz
ð1Þ
zmax d Dz g gt P(z) q u
maximum height (m) thickness of the GVC downdraft shell (m) height step (m) dynamic viscosity (Pa s) heat to work efficiency characteristics of the downdraft shell (kg/m2) density of air (kg/m3) relative humidity of air
Subscripts 0 axis a atmospheric c circular (component) velocity coll collector k kinetic energy p potential energy R radial direction (inward) z component of vertical velocity 1 at radius R1(z) 2 at radius R2(z) Double 00 01 02 11 12
subscripts collector outlet state in the axis at z = 0 state in the axis at z = 10,000 state at radius R1, z = 0 state at radius R1, z = 10,000
Superscripts stagnation state without wc with wz + stagnation state without wc and with wz fr internal friction real state with internal friction 0
Here, mcoll(z) is the adiabatic function of the specific volume of air present in the collector, pa(z) is the surrounding atmospheric pressure at the same height, and g is the gravitational acceleration. The integral (1) is equivalent to the work of the buoyancy force affecting 1 kg of the collector air that has left the collector. In this paper, we refer to integral (1) simply as the air collector ‘‘height potential”. Part of enet coll tech would be used by ground level turbines of a solar chimney power plant with a significantly shorter chimney. The other part of enet coll tech would be used for maintenance of the GVC flow structure and to overcome the internal friction. To provide the optimal case, we will later specify zmax as the upper troposphere boundary: z = zmax = 10,000 m with vertical profiles of pressure and temperature in the atmosphere taken from the standard (NOAA, 1976) atmosphere. This paper proves that, to obtain enet coll tech work by means of vortex structure, heat rejection is necessary after the air
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has passed the turbine. This heat rejection can be realised in two ways: – immediately after passing the turbine, or – at the GVC top. Physical and mathematical models for the latter case are developed later in this paper. 2. Starting assumptions and boundary conditions We suppose that stationary updraft airflow exits a short chimney positioned over the turbines. The updraft has a spiralling upward flow, so that its pressure on the inside is lower than the atmospheric pressure corresponding to the same height. The sub-atmospheric pressure behind the turbines is analogous to the sub-atmospheric pressure maintained in SC-power plants by the chimney effect. The fact that the sub-atmospheric pressure difference inside a GVC would be highest just above the ground means that there is a concentration of air height potential from the collector (1) close to the ground. Hence, the only place where air turbines can be installed is near ground level. The updraft spiralling flow occurs adiabatically with maintenance of angular momentum in only one part of the GVC cross-section, around the radius R1(z) of the central zone. This can be seen in Fig. 1. The goal is to try to avoid the intense internal friction that would occur in the central
zone in the case of strict maintenance of angular momentum. Circular updraft airflow as a solid body is therefore established for R(z) < R1(z) (Fig. 1) by a particular boundary condition. In this respect, the boundary conditions for the GVC are as follows. (A) At the level of the short chimney outlet, where the independent variable z is taken as zero (z = 0), the GVC central section is generated above the central group of turbines for R1 > R > 0 and is taken to be rotating as a solid body and moving upwards with a homogenous vertical velocity w0z(z). For the part of boundary a-b in Fig. 1, we therefore have wz0 ðzÞ ¼ wz0 ð0Þ;
ð2Þ
and for the circular velocity component along the same boundary a–b: wc ð0Þ ¼ wc1 ð0Þ
R ; R1 ð0Þ
ð3Þ
where wc1 ð0Þ is the circular velocity at radius R1(0). The pressure distribution, providing a radial pressure equilibrium for velocity distribution (3), is a known function; it depends on the pressure in the vortex axis p0(0) and rises up to p1(0) at R1(0). It is formulated in Eq. (19) in Section 7. (B) The pressure of the GVC axis p0(z) for 0 < z < zmax is established as a boundary condition along the a–c boundary shown in Fig. 1. This pressure is defined by the vertical isentropic flow of the section of air originating from the
Fig. 1. Two-cell model of the axisymmetrical flow of a gravitational vortex column (with boundary conditions along the broken contour).
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collector. This section of air flow has passed through the central turbine (where pressure is p0(0)) at the short chimney outlet. The connection between p0(0) and p0(zmax) = pa(zmax) depends on the vertical velocity change w0z(z). In the simplest case, when this velocity is either negligible or constant, the function p0(z) simply reflects the hydrostatic relationship: Z zmax q0 ðzÞ g dz; ð4Þ p0 ðzÞ ¼ pa ðzmax Þ þ z
where q0(z) is an isentropic function of pressure p0(z). Generally, p0(z) is a rather complex function that is also dependent upon w0z(z). In any case, the pressure on the GVC axis at the level of the short chimney outlet must be considerably lower than the atmospheric pressure at the same height. (C) The third set of boundary conditions comprises those conditions imposed on the cross-section through which air that has left the main (peripheral) turbines enters the short chimney outlet. Here, the air moves upwards with a radially homogenous vertical velocity 1 w2z ð0Þ, maintaining angular momentum according to the ‘‘free vortex” law. This corresponds to the b–d boundary in Fig. 1, with R2(z) > R(z) > R1(z) for z = 0. We will call this flow or ‘‘layer” within the GVC, which is concentric with the central section, the ‘‘updraft shell”. This flow is established by the given angular momentum a that: wc ð0Þ Rð0Þ ¼ wc1 ð0Þ R1 ð0Þ ¼ const: ¼ a;
ð5Þ
and by the given radial homogenous vertical velocity that: 1
w2z ð0Þ
ð6Þ
for R2 ð0Þ > Rð0Þ > R1 ð0Þ: The radial pressure distribution in this GVC section is defined by (5) and p1(0), in connection with boundary condition A (Fig. 1). The pressure distribution along the b–d boundary section will be formulated in Section 7. (D) The upper GVC boundary is at the top of the troposphere for z = zmax, the c–e contour in Fig. 1; the pressure is atmospheric pa(zmax) = 0.264 bar, R1(zmax) ? 1, R2(zmax) ? 1, and all velocities disappear. The boundary conditions for the horizontal plane at the short chimney outlet, at the axis, and on the top of the troposphere are defined by A, B, C, and D. Model in Fig. 1 represents the vortex column composed by two ‘‘layers”: an updraft central flow and the updraft shell around it. The states of the air in the two layers are different, although both flows belong to the same air collector adiabat. The only remaining undefined boundary condition is that along the outside d–e contour in Fig. 1. Upon first approximation, and from a purely mechanical perspective, these boundary conditions are: pressure is equal to the atmospheric pressure p2(z) = pa(z), and velocities are wc2 ðzÞ and 1 w2z ðzÞ. Mixing and friction with the atmosphere at
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the GVC periphery are treated as negligible in this elementary model. However, the whole process cannot be based on mechanical considerations alone. It must be verified whether the assumed two-cell flow-structure is compatible with the requirements of thermodynamics. 3. Thermodynamic aspects For the vortex column’s peripheral radius R2(0) (point d in Fig. 1), with the boundary condition (5), the pressure p2(0) should be equal to the atmospheric pressure: p2 ð0Þ ¼ pa ð0Þ:
ð7Þ
If the collector air pressure at the collector outlet and the turbine inlet pcoll (point B in Fig. 1) are the same as the atmospheric pressure close to the ground (z = zc), then it holds that: pa ð0Þ ¼ pcoll qa ð0Þ g zc :
ð8Þ
Two very close isobars pa(0) and pcoll = pa(zc) are shown on an h–s diagram in Fig. 2. The state of atmospheric air close to the ground (A) and the state of atmospheric air at the short chimney outlet height (F) are plotted in Fig. 2 (G is the auxiliary state of isentropically expanded ground air up to the same height). Point B is the state of air at the collector outlet, with enthalpy equal to hB and with negligible kinetic energy. If the air did not pass through the turbine, it would have a reduced enthalpy of hC at the chimney outlet, a potential energy of gzc, and stateRC (Fig. 2). This potential energy is F equal to the integral A va dp over the vertical profile in a stable atmosphere (AF in Fig. 2) (see, for example, Ninic (2006)). The potential energy is approximately equal to the isentropic difference in enthalpies hA hG. Hence, the
Fig. 2. Air states between the collector inlet and the short chimney outlet given by the GVC two cell model.
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state at the short chimney outlet – without any work of the turbine and without any gain in kinetic energy – would be C. This state is defined by the line GC, which is equidistant from the isobar AB. The kinetic energy ek, which in this state could be obtained by expansion up to height z = 0, would be hC hE, according to the First Law of Thermodynamics. The pressure at point d in Fig. 1, at the GVC periphery, is pE. Kinetic energy hC hE is gained by the assumption that no work is delivered to the turbine and that the elevation and acceleration process has no internal friction. State D would be the outgoing state of the process with internal friction during elevation/acceleration. Let us evaluate the order of magnitude of this kinetic energy at state E, ek(E), for the short chimney height, zc = 10 m, and for a temperature increase in the collector from 15.0 °C in state A to 45.0 °C in state B. Approximately, this gives hB hC ffi hA hG ffi va ðpcoll pa ð0ÞÞ ffi gzc ¼ 98:1 J=kg hence; pcoll pa ð0Þ ffi 98:1=0:82 ffi 119:0 Pa hB hE ffi vB 119 ffi 0:91 119 ffi 108:0 J=kg hC hE ffi 108:0 98:1 ffi 9:9 ffi ek ðEÞ J=kg; hence the velocity at state E is wðEÞ ffi 4:44 m=s: In the case of zero kinetic energy at the turbine outlet, the relatively small corresponding kinetic energy (9.9 J/ kg) could be replaced by a theoretical amount of work in the turbine. In that case, the total enthalpy after the turbine at radii R2(0) is hE (Fig. 2), and the total work performed in the turbine is hB gzc hE = 9.9 J/kg. Before we make any conclusions based on the previous result, we should discuss an additional issue: can the air at the smaller radii R(0) (i.e., for R2(0) > R(0) > R1(0)) have a total enthalpy lower than hE and provide work greater than 9.9 J/kg? According to the free vortex radial distribution (5) established in the above-mentioned range of radii, it holds that 2 w ð9Þ mdp ¼ d c 2 ; 2
where h0 ð0Þ ¼ hð0Þ þ w2c ð0Þ=2 ¼ const:
ð11Þ
The part of the stagnation enthalpy due to vertical velocity is omitted because, according to (6), this component is constant. Therefore, the stagnation enthalpy h0 does not depend on the radius of the short chimney outlet. This means that all parts of the updraft shell in the radii interval R2(z) > R(z) > R1(z) must deliver the same amount of work to the turbine. Thus, we can conclude that the maximum possible work delivered by the turbine is 9.9 J/kg. Comparing this result for work done in the turbine with the ground-level effect and air velocities in natural tornados, one may conclude that the supposed two-cell GVC model can neither be the basis for a realistically functional GVC device, nor can it form the basis for explaining the destructive power of natural tornados. Now, let us analyse another possible case. Let the pressure behind the turbine at radius R2 be considerably lower than the atmospheric pressure at the same height (p2(0) < pa(0)). Such a state corresponds to point H in Fig. 3, instead of points E in Figs. 2 and 3. In Fig. 3, the potential energy increase up to the chimney outlet is gzc, the shaft work thus obtained in the turbine is wt, and the kinetic energy behind the turbine is ek. This setup allows for the production of considerably more work and kinetic energy, but with one condition: the air passing through the turbine must be able to attain atmospheric pressure at the R2(0) radius (state J in Fig. 3). Before that, this air must reject a certain quantity of heat, Dq in Fig. 3. The reason for this is the same as in the case of condensate return in the steam boiler after heat rejection occurs in the condenser. More sophistically, this is due to the divergence of isobars in the h–s diagram, as a consequence of the Second Law of Thermodynamics. State J could be achieved by
with 1 w2z ð0Þ ¼ const. Relation (9) means that all air states in this cross section belong to the same isentrope (passing through B in Fig. 2). Therefore, according to the First Law of Thermodynamics, in this range of radii we have mdp ¼ dh:
ð10Þ
Hence, from (9) and (10), it follows that the specific stagnation enthalpy h0 is constant over the same cross section: dh þ dðw2c =2Þ ¼ dh0 ¼ 0; 2
See Section 7.
Fig. 3. The states of air between the collector outlet B and the short chimney outlet H.
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the transformation of kinetic energy (ek(I)(=ek(H)) into pressure increase p(J) p(I)). Based on this discussion and on the fact that in nature there exists quasi-stationary tornado funnels with very destructive effects at ground level, we can conclude that the existence of a GVC column with a near ground concentration of availability (for turbine work) is possible, because of the fact that it exists in nature. However, it cannot be modelled by a two–cell vortex without heat rejection. In contrast, according to Fig. 3, with heat rejection it should be p2 ð0Þ ¼ pðHÞ < pa ð0Þ:
ð12Þ
Therefore, a considerable supplementation of the two-cell model is necessary in order to bridge the pressure ‘‘gap”, pa(0) p2(0). There are only two possible ways of achieving this, one of which is proposed and developed here. 4. Physical model formulation According to the previous section, after adiabatic expansion in the peripheral turbine (in Fig. 1), the collector
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air can only regain its high initial pressure (atmospheric pressure at close to ground level) if it previously rejects a sufficient quantity of heat. The first way to achieve this is by filling the zone around radius R2 with high pressure airflow (pa > p > p2). This airflow is provided from air that has passed the peripheral turbines and reached the top of the GVC, where it rejects the heat. After heat rejection, cooled air descends toward the ground, rotating around the updraft flow at radii R2(z) (flow E in Fig. 4). We can suppose that this descending airflow would have the form of a rotating annular shell of thickness d(z), which would embrace the updraft shell. This physical model is named ‘‘three-layer” model. On the other hand, cooled air descending from height zmax in the standard stable atmosphere could not descend without containing a large amount of condensate or ice droplets in its volume. It would be a downdraft induced by falling precipitation. Adiabatic saturation with steam evaporated from condensate droplets increases the cooled air density and contributes to its descent. Condensate droplets in the descending shell arrive in the descending shell via
Fig. 4. A simplified three-layer physical model of the GVC stationary vortex column.
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the updraft shell. The descending shell, due to mixing with the atmosphere, has a specific angular momentum lower than the initial one (a0 6 a). The external shell movement can therefore be assumed to be circular and downward (see Fig. 4), yet, overall, it is considerably more complex. 4.1. Second mechanism for pressure gap over-bridging There is another way of rejecting heat from the air after it passes through the turbines and regains atmospheric pressure. This consists of bringing the air at the turbine outlet into contact with cold, low-pressure external air and allowing the two to mix. This can be achieved via a weakly forced central downdraft of cold, high-level atmospheric air. These air parcels would stop and mix with collector air over the turbine, thus allowing the collector air to obtain atmospheric pressure (Ninic and Nizetic, 2007). This model of a GVC is named the ‘‘central downdraft model”. Some sources, for example Bluestein (2005), have indicated that such contact with high, cold air occurs at a certain low altitude above the ground in the central zone of natural tornados and ‘‘dust devils”. The effect of cooling by the central descending airflow could be increased and/or initiated by injecting liquid water into the space behind the turbine. 4.2. Additional validation of the physical model with a downdraft annular shell In addition to the thermodynamic validation of the three-layer model (Fig. 4), we can also check the idea by using existing numerical models of quasi-stationary tornadoes, such as those in the work of Markowski et al. (2003). In this model, which is inspired by observations and experimental readings made in the field with natural tornadoes, the authors concluded that a descending airflow (in the form of a rotating shell) is necessary for tornado development to quasi-stationary states. In this paper, as in our three-layer model, the radial pressure drop corresponds to the centripetal acceleration in the rotating downdraft shell. We also note that, in consideration of tornado genesis, the emphasis is on the important role of the descending and rotating annular shell, as well as on the transmission of angular momentum a0 . This angular momentum is initially at its maximum at the top of the rotating annular shell and at approximately zero near ground level. However, our proposed three-layer GVC model does not have solely tornado-specific characteristics, and the genesis of GVC is not analogous to that of tornadoes. In Bluestein (2005) and Snow and Randal (1984), tornadoes and dust devils were classified as ‘‘one celled” or ‘‘two celled.” The ‘‘two celled” types are in good agreement with the GVC ‘‘central downdraft model”. Our analysis provides conclusions that are in agreement with the observations of natural vortex structures, including the bridging
Table 1 An explanation of the labels used in Fig. 4 Label
Label description
A T1 T2
Collector roof Annular space for peripheral turbines with electric generators Turbines defining flow B in the GVC central part, rotating as a solid Flow of ascending vortex air shell leaving the peripheral turbines Heat sink, i.e. the space in which collector air delivers heat to the environment ‘‘Black box” – descending spiralling shell
C D E
of the pressure gap in the ‘‘three-layer model” (with a downdraft outer shell). Furthermore, we developed the simplified the threelayer GVC model based on the following propositions: – Pressure ‘‘gap” filling is achieved by the externally rotating descending shell. – For the GVC to reach the top of the troposphere, the appropriate radial pressure equilibrium condition for a rotating descending shell (thickness, effective density, and circular velocity) should be satisfied. The descending shell structure could be composed of a mixture of saturated air and liquid or ice droplets. The complex structure of the shell and its internal motion is not described in the proposed three-layer physical model. Descending shell volume is a boundary condition and is in the form of a ‘‘black box” that serves to fill the pressure gap between the GVC upflow and the atmosphere. Vertical and cross section views of such a GVC model are shown in Fig. 4, with explanation of the labels given in Table 1. Fig. 5 is a schematic of the vertical section of one-half of a GVC, with boundary conditions underlined. These conditions are defined at the level of the short chimney outlet (z = 0), along the axis and at the top level of the troposphere (z = 10,000 m). Side contact with the atmosphere is shown as a band and not a line. This band is a descending rotating shell with an unknown internal structure. This structure is characterized by the function P(z), which determines a mechanical radial equilibrium with the atmosphere. This function is analytically defined in Section 7. 5. Internal friction and dissipation of angular momentum Fig. 4 shows profiles of three axially symmetric layers that are characterized by their pressure (p), circular velocity (wc), and vertical velocity (wz) distributions in the cross section of the simplified GVC model. The rising GVC flow, 0 < R(z) < R1(z), rotates as a solid body (a ‘‘forced vortex” in Rankine’s terminology). According to Rankine’s ‘‘free vortex law”, the annular rising flow, (R1(z) < R(z) < R2(z)), is the section of spiral air movement from the peripheral turbines. The third layer belongs to the descending condensate and cooled air shell, which also has a circular velocity component. The linear distribution of circular velocities in
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of radius R1. The stationary free vortex flow is such that the moment of the friction force, by which each fluid layer affects the adjacent one, is independent of the radius. The viscous force moment per unit height of the vortex equals (Binder, 1958): M ¼ 4pgwc R:
ð13Þ
Due to the independence of this expression from the radius, it follows that wc R ¼ const: ¼ a; which is the ‘‘free vortex law”, as in our model updraft shell. The same angular momentum M would be lost per second at the outer cylindrical surface (R = R2) of such a flow with viscous force momentum. Suppose that this dissipation of angular momentum is compensated by a centripetal radial flow without angular momentum. The flow per unit height is m_ R . This yields 4pgwc R ¼ m_ R wc R ¼ qwR 2pRwc R; where wR ¼ Fig. 5. Boundary conditions of the GVC section.
the central rotating flow is not established spontaneously by friction, but rather is imposed at the outlets from the central turbines. The primary source of internal friction in annular rising flow is associated with differences in vertical velocities at the radius R2(z). This includes the secondary effect of ‘‘heat addition”, analogous to the ‘‘heat return” effect found in multistage turbine processes. Beyond the loss of availability, friction on boundary surfaces may also result in an outward radial transfer, as well as a radial loss of angular momentum in the direction of a static atmosphere. The neglect of the radial dissipation of angular momentum can be justified by one of the following reasons (or by a combination of them). The first possible reason is a low intensity of radial angular momentum loss per square meter in the rotating flow. The second reason for the relatively small angular momentum dissipation could be the small ratio between the perimeter and the cross sectional area of the corresponding GVC flow section. Finally, a third reason for such a small dissipation could be the existence of at least a small radial air inflow, which is sufficient to compensate for most of the radial angular momentum dissipation by friction. For a rough numerical estimate of these arguments, let us first examine the entire laminar flow of a viscous fluid extended to infinity and moving around a rotating cylinder
2g qR
ð14Þ
is the centripetal radial velocity of the surrounding air by which the loss of angular momentum in the environment would be fully compensated. For a numerical estimate of wR, we anticipate the dimensions and velocities in one characteristic GVC with R2 = 15 m and wc2 ¼ 60 m=s for g = 106 Pa s. Hence, from (13) and (14), M = 0.0113 N/m, m_ R R2 wc2 ¼ 14; 130; 000 kg m=s2 , and wR = 0.00000013 m/s. Therefore, in this case, the radial inflow necessary to compensate for the momentum radial dissipation is negligible in relation to the corresponding overall angular momentum flow. The phenomenon of instabilities in annular rotating upflows enveloped by rotating downdrafts has not been well studied in the literature; this is a subject in need of additional research effort. Because of this, we can only rely on the previous works of Taylor and Prandtl, elaborated in Schlichting (1960). In these works, the limits of stability were defined in the case of laminar flow, but for circumstances different than those in our case. Their work is related to an annular rotating fluid flow which is set between a rotating cylinder of radius R1 and a cylinder at rest (which has an appreciably greater radius). The situation here, with the annular space of the rotating fluid lying between the sections of the updraft annular flow and the downdraft annular flow (together from Rmin = R1 to Rmax = R2 + d) is relatively well described by such circumstances. According to Schlichting (1960), we can conclude that instability appears according to the following criterion:
470
wc1 ðR2 R1 þ dÞ m
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 R1 þ d > 41:3: R1
ð15Þ
This instability is caused by a centrifugal force that is higher on the fluid particles near the inner radius R1. Instability results in the appearance of cellular vortices with axes directed along the circumference. Anticipating the dimensions and velocities in the same numerical example, we can conclude that the dimensionless left side of the inequality criterion is rffiffiffiffiffi 60 10 10 ffi 36 1015 : 15 1015 12 This is far beyond the stability limit value, and it even stretches considerably into the domain of turbulent flow. Closed streamlines of secondary motions (caused by instability), however, are fully inside the space of free vortex flow and have a homogeneous radial distribution of angular momentum. In this case, intensive internal friction in the updraft’s annular flow does exist, but the radial flow of angular momentum is not very significant. These conclusions are supported by some well-known observations made on natural tornado ‘‘funnels” that have been published in the Encyclopaedia of Science Technology (1978). The first such observation is that quasi-stationary tornadoes may exist for several hours without a renewable source of angular momentum. The second observation regards the way in which a tornado funnel disappears; it does not disperse radially in the atmosphere, but rather becomes more slender, until it takes the form of a rope that eventually breaks. Overall, this supports the hypothesis that there is little outward loss of angular momentum through the surface R2(z). Furthermore, we may conclude that the downdraft shell (with R(z) > R2(z)) has a structure that is more complex than that of the updraft shell. In this simplified model, the downdraft shell is treated as a ‘‘black box” that serves to fill the pressure gap: pa(z) > p(z) > p2(z). In neglecting the radial loss of angular momentum, we do not neglect the internal friction caused by the viscosity of the fluid and by different forms of motional instability. However, in this simplified analysis, the internal friction Dwfr in the energy equation written for the stream line along the annular updraft flow3,
In conclusion, we can say that, in this paper, the internal friction is still taken into account, but the unused part of availability at the top of the GVC has been purposely left out. The influence of internal friction can be investigated by varying the unused quantity of availability at the top of the GVC. 6. Additional boundary conditions In this section, we complete the boundary conditions defined in Section 2. Let us start from the section describing a short chimney outlet on the ground level, z = 0. The radius R1(0) is chosen to be of sufficient magnitude such that circular movement of air with constant angular momentum is provided for R(0) > R1(0). For R2(0) > R(0) > R1(0), relation (5) is satisfied. Here, the pressure distribution along the radius at z = 0 is such that radial equilibrium is fulfilled. Accordingly, the pressure gradient is proportional to the centripetal acceleration of circling fluid particles: dp ¼ qw2c
dR ; R
ð16Þ
with (5): w3c adwc 2 ¼ wc dwc ; mdp ¼ a wc vdp ¼ wc dwc ; which is anticipated by the previously stated relation (9). According to the First Law of Thermodynamics for reversible fluid flow: dq ¼ dh vdp:
ð17Þ
The imagined reversible adiabatic movement of a fluid parcel along the cross section in updraft shell is obtained by dh ¼ mdp ¼ wc dwc ; hence, h þ w2c =2 ¼ h0 ¼ const:;
ð18Þ
0
where h is the corresponding stagnation enthalpy. The vertical component of the velocity 1 w2z ð0Þ is homogenous over the cross section at a given radius and contributes to the total stagnation enthalpy h+:
vDp ¼ Dep þ Dek þ Dwfr ;
hþ ¼ h þ ðw2c þ w2z Þ=2 ¼ const:;
does not need to be expressed explicitly (as Dep and Dek). The process along the streamline will be treated by neglecting internal friction (without Dwfr4), but with a requirement that a certain positive value of availability in the annular rising flow at the top of the GVC is retained. By accomplishing this, in this simplified model we have ‘‘reserved” a place for Dwfr in the simplified energy Eq. (21).
as a property independent of radius R, for R1(0) < R(0) < R2(0). Based on the previous equations, and according to Fig. 5, the completed boundary conditions are the following. (A) At the level of all turbine outlets, the radius R1(0) and vertical velocities wz(0) for both the central and the peripheral turbines (more precisely for the short chimney outlet, at), are given in Figs. 4 and 5. The pressure gap pa(0) p2(0) is also a feature of the boundary conditions connected with downdraft shell. Pressure, temperature,
3 4
See Eq. (210 ). See Eq. (21).
ð180 Þ
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mixing ratio x, and mass flow rates are all given at the collector outlet. The mass flow rates are: m_ 01 through section 0–1, m_ 12 through section 1–2, and the overall mass flow rate is m_ ¼ m_ 01 þ m_ 12 . These data also define the profiles of all air properties at height z = 0. (B) Temperature, pressure, and density profiles along the GVC axis are imposed by the vertical airflow from the central axial turbine. This means that the pressure on the axis at 10,000 m is the atmospheric pressure at that height, and that other properties along the axis are determined by the reversible adiabat of the upward moving collector air. For the simplest case, the distribution of pressure p0(z) corresponding to constant vertical velocity wz0 is shown in Fig. 6. This shows the stepwise rise of pressure near the top of the troposphere due to the transformation of kinetic energy from ðwz0 ðzmax ÞÞ2 =2 to Dp/q (as explained in Section 2, paragraph B). In principle, taking into account the variation of wZ 0 ðzÞ with height does not cause complications, except for the additional imposition of continuity and the First Law of Thermodynamics for axial flow. In any case, whether constant or variable, the velocity wz0 ðzÞ plays a double role: 1. To ensure the presence of the central turbine air flow in the axis of GVC. 2. To ‘‘translate” the static distribution of pressure p0(z) 2 towards lower values for q½wz0 ðzÞ =2. For an illustration of this, compare the full line p0(z) and the broken line shown in Fig. 6. (C) There is a (standard) atmospheric pressure profile on the outer side of the downdraft shell. In this model, the effect of the downdraft shell on the GVC updraft shell is only reduced to radial pressure difference. (D) At height zmax = 10,000 m, the environmental temperature is 50.0 °C, the pressure is 0.264 bar, and practi-
cally all moisture is in the form of small ice crystals. We assume that the air forming the downdraft shell has an initial pressure and temperature equal to the corresponding atmospheric pressure and temperature, containing additionally all the moisture in the form of condensate. As such, the downdraft descends adiabatically, including a constant saturation process due to the presence of the condensate. Saturation by vapour and the presence of a sufficient quantity of condensate are prerequisites for this air to descend in a stable atmosphere. 7. Analytical formulation In the analytical articulation of the physical model presented here, in proceedings of paper we will try to ensure that the relations used are not of speculative nature, with potential applications only in particular circumstances. Rather, we will try to ensure that they are based on general principles of mass, momentum, and conservation of angular momentum, the First Law of Thermodynamics for various observers (from various frames of reference), and the Second Law of Thermodynamics. Additionally, the equations used to describe humid air states in a psychrometric chart will be assumed to be applied throughout. First, we summarize the unknown functions of height (z), on which any geometrical or physical properties of any GVC cross section may depend. With the given boundary conditions, such properties are fully defined in terms of five unknown functions of height: R1(z), p1(z), 1 w2z ðzÞ, R2(z), and p2(z). The radial pressure distribution and circular velocities are defined by R1(z), p1(z), a and p0(z), within the interval 0 < R(z) < R1(z). The vertical velocity in this interval is homogenous, approximately constant with regard to height, and equal to 0 w1z ð0Þ. The temperature and density distributions within the same interval 0 < R(z) < R1(z) are also defined by the fact that all states of air originating from the collector belong to the same isentrope. The distribution of all properties within the interval R1(z) < R(z) < R2(z) is defined by R2(z) and p2(z), including circular velocities, while the vertical velocity is homogenous and equal to 1 w2z ðzÞ. The five listed unknown functions at height z + Dz are derived from their values at the previous level z, from five independent equations, and from the side boundary conditions. All five equations are conservation laws of general significance. The first equation is the radial pressure equilibrium condition (17) applied within the radius interval 0 < R(z) < R1(z) to fluid rotating as a solid body (according to law (3)). By integration, we get p1 ðzÞ p0 ðzÞ ¼
Fig. 6. Illustration of axial pressure distribution along an axis and the effect of wz0 disappearing near the top of the troposphere.
471
10 1 q ðzÞw2c1 ðzÞ; 2
ð19Þ
where 0 q1 ðzÞ is the corresponding average density of the fluid. The second relation is the conservation of the mass flow rate for the updraft shell:
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df1 w2 ðzÞ1 q2 ðzÞ½R22 ðzÞ R21 ðzÞg ¼ 0:
ð20Þ
The third relation is derived by applying the First Law of Thermodynamics from the view of two different thermodynamic observers. In its general form, it is5: v1 ðzÞdp1 ðzÞ ¼ dep1 þ dek1 þ dwfr ¼ gdz þ w1 ðzÞdw1 ðzÞ þ dwfr :
ð210 Þ
Here, w1(z) is a module of the velocity vector at radius R1 ðzÞðw21 ¼ w2c1 þ w2z Þ, and dwfr(P0) is the work due to internal friction. Neglecting this friction work for the time being, we get m1 ðzÞdp1 ðzÞ ¼ gdz þ 1 w21 ðzÞdw1 ðzÞ:
ð21Þ
The motivation for neglecting friction work was explained at the end of Section 5. The fourth independent relation is the radial equilibrium condition for the fluid inside the updraft shell (17); integrating (17) with (16) gives p2 ðzÞ p1 ðzÞ ¼
11 2 q ðzÞ½w2c1 ðzÞ w2c2 ðzÞ: 2
ð22Þ
Finally, the radial pressure equilibrium condition for the downdraft shell (modelled as a”black box”), also according to (17), is pa ðzÞ p2 ðzÞ ¼ ðqdÞðzÞ
w2c ; RðzÞ
ð23Þ
where qd(z) is the product qd as a function of height. Here, pa(z) is the atmospheric pressure at height z; q and d are the effective density and downdraft shell thickness, wc ðzÞ is the effective circular velocity, and RðzÞ is the average radius of this shell. Eq. (23) can be specified by pa ðzÞ p2 ðzÞ ¼ PðzÞ
w2c2 ; R2 ðzÞ
ð24Þ
where PðzÞ ¼ ðqdÞ
w2c w2c2
R2 ðzÞ : RðzÞ
upon which the whole GVC structure depends. This includes its state at the short ‘‘chimney” outlet section. In the following sections, we present a detailed numerical solution for a particular GVC chosen as characteristic of many solutions with different adopted constructive parameters, plant parameters, and corresponding height stratification of the values of the P(z) functions. The adopted atmosphere was standard in all cases (NOAA, 1976), its maximum height being 10,000 m. Atmospheric temperature, pressure, and density at the ground level are 15.0 °C, 1.0132 bar, and 1.225 kg/m3; at the height of 10,000 m, temperature, pressure, and density are 50.0 °C, 0.26436 bar, and 0.413 kg/m3, respectively. Ideal gas relations are used in the thermodynamic calculations of moist air properties. According to ASHRAE (2005) (section: Thermodynamic properties of moist air), errors in enthalpy and specific volume calculations at standard atmospheric pressure for a temperature range of 50 °C to 50 °C are less than 0.7%. These errors decrease with a decrease in pressure. 8. An algorithm and characteristic numerical solution For the numerical calculations, our algorithm starts from the first numerical step of the Dz increment and continues with the assumed, relatively low vertical velocity of the central flow (this is assumed on the axis as well). The pressure effect of a vanishing low vertical velocity is only added in during the last step of the algorithm. Vanishing of the velocity is connected by the abrupt extension of the radius R1(z) immediately before z = 10,000 m. The initial parameters and boundary conditions are shown in Table 2 (all states of air in Table 2 are marked according to the notation in Fig. 7). Qualitatively, this and other states are shown in the p–m diagram in Fig. 7. Values for Table 2 Initial parameters and boundary conditions
ð25Þ
The P(z)-function functions in boundary condition C, Section 6, which specifies the downdraft shell as a ‘‘black box”. The downdraft shell is separated from the rest of our simplified model, as an independent boundary condition. The relations established by (19), (20), (21), (22) and (24) compose a system of five equations with five listed unknown functions of height. The function P(z) in Eq. (24) is not an unknown function but a boundary condition 5 We note in passing that the relation (210 ) can be obtained from the rate of change of momentum multiplied by the fluid parcel vertical displacement. Thermodynamic derivation (210 ), emphasized here, is more general and based on the First Law of Thermodynamics for two different observers: one is immobile with respect to the parcel in motion, while the other is immobile with respect to the Earth’s surface. The connection between the mechanical and thermodynamic aspects of Eq. (210 ) is discussed in more detail in Ninic (2008).
Ground-level state of standard atmosphere, denoted by ‘‘1a” t1a(0) = 15.0 °C; pa1(0) = 1.012 bar; q1a(0) = 1.225 kg/m3; pa(10,000) = 0.26436 bar h1a = 15,075 J/kg (enthalpy of dry air) Air state at the collector outlet denoted by ‘‘00” t00(0) = 45.0 °C; p00(0) = 1.012 bar; x00(0) = 0.0182 kgd./kgd.a.; h00 = 92,192 J/kgd.a.; s00 = 320.4 J/kgd.a. K Air state at the central turbine outlet (in the axis od short chimney), denoted by ‘‘01” t01(0) = 34.9 °C; p01(0) = 0.90818 bar; wz01 = 10.0 m/s; h01 = 82,351 J/ kg; R1(0) = 10.2 m 2 hþ 01 ¼ 82; 351 J=kg; wc1 = 58.8 m/s (a = 600 m /s). Air state at the peripheral turbine outlet in R1(0) denoted by ‘‘11” t11(0) = 36.7 °C; p11(0) = 0.92592 bar; wz11 = 30.0 m/s; h11 = 84,017 J/ kg; hþ 01 ¼ 86; 197J=kg Downdraft shell as the ‘‘black box” boundary condition P(z) P(0) = 45.7 kg/m2
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473
wt0 ¼ h00 hþ 01 ¼ 92; 192 84; 274:3 ¼ 7917:7 J=kg; where the average enthalpy is defined by þ hþ ¼ ðh þ h Þ=2:0 ¼ 84; 274:3 J=kg. The specific shaft 01 01 01 work in the peripheral turbines is wt1 ¼ h00 hþ 11 ¼ 92; 192 86; 197:6 ¼ 5994:4 J=kg:
Fig. 7. p–v diagram of atmospheric states and processes in the GVC.
enthalpy and entropy of moist air are declared to be zero according to ASHRAE (2005) for dry air and liquid water at p = 1.012 bar, t = 0.0 °C. In the presented numerical example, properties of the air state at the turbine inlet (00) (Fig. 7) are: p00 = 1.012 bar, h00 = 92,192 J/kg, s00 = 320.4 J/kgK, and t00 = 45.0 °C. The shaft work done in the central turbine is
The state 01 also dictates the whole boundary condition along the axis: p0(10,000) = 0.264 bar = pa(10,000). Other boundary conditions in this numerical example are R2(0) = 12.73 m, wc2 ð0Þ ¼ 47:1 m=s, p2(0) = 0.93236 bar, and mass flow rates m_ 01 ¼ 3314:6 kg=s and m_ 12 ¼ 5685:4 kg=s. The results are reviewed in Table 3, which gives the values of the main parameters of the process, including the heat received in the collector, the heat to work efficiency, and total turbine power. In this numerical example, the function P(z) was chosen as a boundary condition because it is compatible with the constant vertical velocity component in the central flow 0 w1z and with the mass conservation law. Additionally, the zero height in the numerical program, and in Table 3 and Figs. 8 and 9, refers to ground level, and not to the short chimney outlet height. The acquired GVC geometry is illustrated in Fig. 8. In Fig. 9, a more detailed GVC section at the height of 4000 m is shown, along with radial distributions of characteristic properties.
Table 3 State values by height Boundary conditions (m) a (m2/s) Rstart 1
w0z 1 (m/s)
w12 zp (m/s)
m (kg/s)
P[0] (kg/m2)
10.20
10.00
30.00
9000.0
45.71
Turbines and down-stream parameters h01 (J/kg) hþ h00 (J/kg) 01 (J/kg)
h11 (J/kg)
hþ 11 (J/kg)
h02 (J/kg)
hþ 12 (J/kg)
92192.0
82301.0
82351.0
84017.5
86197 6
15326.7
11804.2
wtc (J/kg)
wtp (J/kg)
Qd (MW)
gt (%)
P(MW)
7917.7
5994.42
694.053
8.7
60.325
Simulation results z (m) R1 (m)
R2 (m)
p1 (bar)
p2 (bar)
pa (bar)
w01 z (m/s)
w12 z (m/s)
wc1 (m/s)
wc2 (m/s)
P (kg/m2)
10.0 100.0 300.0 600.0 900.0 1000.0 1500.0 2000.0 3000.0 4000.0 5000.0 6000.0 7000.0 8000.0 8000.0 10000.0
12.73 12.66 12.68 12.74 13.28 13.29 13.37 13.58 14.09 14.69 15.33 16.02 16.75 17.53 18.34 Inf.
0.92592 0.91637 0.89619 0.86655 0.83767 0.82821 0.78177 0.73744 0.65559 0.58133 0.51419 0.45365 0.39914 0.35010 0.30602 0.26441
0.93236 0.92216 0.90165 0.87156 0.84301 0.83339 0.78594 0.74085 0.65826 0.58342 0.51582 0.45491 0.39998 0.35085 0.30659 0.26441
1.01205 1.00129 0.97773 0.94322 0.90970 0.89875 0.84556 0.79495 0.70108 0.61640 0.54020 0.47181 0.41061 0.35600 0.30742 0.26441
10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10 00 10.00 00.00
30.00 32.82 34.13 35.91 30.19 30.78 35.45 39.27 42.29 45.09 48.04 51.29 54.88 58.79 63.00 00.00
58.82 58.02 57.54 56.82 56.08 55.83 54.05 52.22 49.80 47.41 45.08 42.83 40.67 38.60 36.64 00.00
47.11 47.36 47.28 47.08 46.16 45.12 44 86 44 17 42.55 40.84 39.12 37.43 35.80 34.22 32.69 00.00
45.71 44.70 43.10 41.20 43.40 42.70 39.60 37.61 33.31 29.01 24 41 19.21 13.81 7.61 1.31 00.00
6000
10.20 10.33 10.42 10.55 10.69 10 74 11.10 11.48 12.04 12.65 13.30 14.00 14.75 15.54 16.37 Inf.
GVC simulator design by Sandro Nizetic.
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Fig. 8. A characteristic GVC cross section.
9. Discussion of numerical results The characteristic air states of a standard atmosphere at various heights are illustrated in Fig. 7 (curve 1a 2a). The collector air state at the collector outlet is denoted by 00, while the state at the peripheral turbine outlet is 11 (in radius R1 at the short ‘‘chimney” outlet). The state at the central turbine outlet is 01, and the state at the top of the troposphere is marked as 02 for the central turbine and 12real for the peripheral turbines. Stagnation states corresponding to the particular states are marked by a superscript +. The total enthalpy difference between the air from the central and peripheral turbines at the height of 10,000 m, þ hreal 12 h02 ¼ h12 h02 (shown in the p–v diagram), is the work capacity conserved during the isentropic process of the air passing through the peripheral turbines (particularly with R = R1(z)). This travels all the way to the top of the troposphere. At a height of 10,000 m, it takes the form of kinetic energy hþ 12 h02 . In reality, this work capacity would be consumed by internal friction during the elevation process in the updraft annular flow, according to the last passage in Section 5. However, according to that section, a distributed internal friction process is substituted for throttling 12+–12real at the top of GVC (see Fig. 7). The broken curve 00–12real represents the real process, with regularly distributed internal friction. The real state of the air from peripheral turbines is at the top of the troposphere 12real, defined by the equality of enthalpies in states 12+ and 12real.
Fig. 9. A characteristic GVC axial section, z = 4000 m.
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The air that has passed through the central turbine has obtained state 02 at z = zmax, having lost its low velocity. In this way, the mean state of the air at the top of the GVC is the result of an adiabatic and isobaric mixing of the flows of states 02 and 12real with the corresponding mass flow rates. After heat rejection at z = zmax, the difference between the air flows from different turbines disappears (therefore, there is no more need for double indices). In Fig. 7, the elements of energy transformation are thus represented in general form. The area 00p –1a –2a –12real p represents the potential energy at the top, equal to Dep = gzmax = 98,100 J/kg. Curve 00–02 shows the reversible process of the collector air starting with state 00, going to the state it would occupy at the top of the troposphere after passing through the central turbine. The area 1a–00–02– 2a, therefore, represents the height potential of the collector air as defined by (1). The state behind the peripheral turbine (or turbines) in radius R1(0) is marked by 11, and the air stagnation state is marked by 11+. According to this, the þ area 11þ p –11 –11–11p is the kinetic energy in state 11, and the area 00–11þ –11þ p –00p is the specific work done in the peripheral turbine. The process is analogous for the central turbine, with the state behind it being 01, i.e., 01+. As previously stated, the state of the updraft shell air at the top of the troposphere with internal friction is 12real. The internal friction work wfr1 would, in such a case, be þ real –12real equal to the area 12þ p –12 –12 p , and the work lost þ real due to friction is equal to the area 12þ p –12 –02–12p . As known from detailed analyses of flow processes, the internal friction work has a double effect: work loss Dwfr and heat gain are the same amount. This ‘‘heat”, however, has a secondary favourable influence on the work gained, so that the loss of work due to friction is somewhat smaller than the work of friction itself. We will use the h–s diagram to enter the actual data of characteristic states and for a presentation of the process (see Fig. 10). 00–11+ is the stagnation state change in the peripheral turbines at radius R1(0). The process performed by the collector air that has passed through the central turbine on the GVC axis is 00–01(ffi01+); the state of the air that has left the central turbine is described by pressure p01 = 0.90818 bar, enthalpy h01 = 82,301 J/kgd.a., temperature t01 = 34.9 °C, and kinetic energy ek01 = 50 J/kgd.a. In state 02 at the top of the troposphere, the pressure is p02 = 0.26436 bar (=pa(10,000)), h02 = 15,327 J/ kgd.a., and t02 = 17.4 °C. The state behind the peripheral turbines, at radii R1, is described by stagnant properties þ and tþ pþ 12 ¼ 0:27789 bar; h12 ¼ 11; 804:2 J=kgd:a: , 12 ¼ 15:3 C. The rise in entropy due to internal friction conis Dsfr ¼ s12real sþ centrated at zmax 12 ¼ 13:8 J= real kgd:a: K; s12 ¼ 320:4 þ 13:8 ¼ 334:2 J=kgd:a: K. The work lost due to friction is h12þ h02 ¼ 3523 J=kgd:a: . After heat rejection and complete equilibrium with the environment is attained, the air state is 3, as shown in Fig. 10. The air here is saturated at t3 = 50.0 °C, with negligible steam content (ffi0.1 g/kgd.a.) and with all 18.2 g/kgd.a. of water content in the form of ice. The prop-
475
Fig. 10. Mollier h–s diagram of the process in GVC and the Brayton cycle.
erties are h3 = 57,941.5 J/kgd.a. and s3 = 149.8 J/kgd.a. K. Lowered isentropically to ground level, the stagnation enthalpy of this air would be increased by the amount of potential energy (98,100 J/kg) and would, therefore, be equal to 40,158.5 J/kg. At an atmospheric pressure of 1.012 bar, saturated air at 14.5 °C has an enthalpy approximately equal to 40,160 J/kg. Its gas phase remains at about 8.0 g/kgd.a., with all liquid water in droplet form. Overall, we reach state 4, as shown in Fig. 10, with h4 = 40,158.5 J/kg. There, atmospheric air has an enthalpy of only about 15,075 J/kgd.a at ground level. The moist air in the GVC actually performs the part 00– 12real (02)–3 of the Brayton cycle in the gravitational field (Fig. 10). Internal friction in the updraft shell in process 00–12real decreases the specific work of the peripheral turbine and therefore has an important role in the structure of the GVC. A realistic cycle is much more complex than the Brayton cycle in Fig. 10. The real process of air down-drafting (after the heat rejection at a level of zmax ffi 10,000 m) is different than the one presented as the broken curves 3–4. The real state of this air after sinking to ground level is that of standard atmosphere at the collector inlet (regardless of whether all the air is used in the downdraft shell or not). The real process that leads to the standard atmospheric state at the inlet of collector is not of a local, but of a regional or even global kind. Namely, the air of state 3 (Fig. 10) is thrown into the atmosphere (as a huge reactor) on the level zmax. From the other side, the air is taken to be at
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atmospheric state at ground level. The overall heat to work efficiency is, for this reason, defined as the ratio between the work extracted from the turbines and the enthalpy difference in the collector (coming as added heat). A numerical example leads to a resultant turbine power of approximately: P ¼ 3315 7918 þ 5685 5994 ffi 60; 324:57 W ffi 60 MW: The heat to work efficiency with a standard atmospheric state at the collector inlet is gt ¼ ½P =ð9000 ð92; 192 15; 075ÞÞ 100 ¼ 8:7%: 10. Conclusion This paper begins with the technically useful part of heated moist air availability in the atmosphere, stratified in a gravitational field. Relying on other authors’ work (cited in our reference list), we accepted and elaborated on the possibility of a concentration of availability at ground level without a concrete chimney. Our proposed model’s main contribution relates to a thermodynamic approach that introduces a restriction of air return to the atmospheric stagnant pressure. Two possible solutions for solving the previously mentioned restriction are specified herein. One of the solutions is elaborated on, and it results in a simplified mathematical model of the process. This process is partially conducted in the solar collector and in ground level turbines. It is also vertically conducted through the troposphere as a flow in a gravitational vortex column (GVC). The rest of the process in the elaborated solution is connected with the heat rejection at the top of the troposphere and with power cycle closing. Typical integration results are shown by one characteristic numerical example of a GVC (as a part of a cycle) in the form of a table as well as in the form of a Mollier h–s diagram. Acknowledgements We would like to thank the anonymous reviewers and the Chief Editor for useful suggestions and recommendations. We would also like to thank the Croatian Ministry of Science, Education and Sports for funding this project. References Bernardes, M.A.S., Voß, A., Weinrebe, G., 2003. Thermal and technical analyses of solar chimneys. Solar Energy 75, 511–524.
Binder, R.C., 1958. In: Advanced Fluid Mechanics, Vol. II. Pretince-Hall, New Jersey, pp. 52–54. Bilgen, E., Rehault, J., 2005. Solar chimney power plants for high latitudes. Solar Energy 79, 449–458. Bluestein, H.B., 2005. A review of ground-based, mobile, W-band Doppler-radar observations of tornadoes and dust devils. Dynamics of Atmospheres and Oceans 40, 163–188. Dessens, J., 1962. Man-made tornadoes. Nature 193, 13–14. Dessens, J., 1969. Etude des vortex du type tornado a´ partir de modeles de laboratorie. Thesis, University of Paris. Dessoliers, H., 1913. Comment l’homme accroirta progressivement les pluies des re´gions arides. Refoulement du Sahara, Alger. Group of Authors, 1978. Encyclopedia of Science & Technology 18. McGraw Hill, New York. Group of Authors, 2005. ASHRAE Fundamentals Handbook. ASHRAE, Atlanta. Haff, W., 1984. Solar chimneys, part II: preliminary test results from the Manzanares pilot plant. International Journal of Solar Energy 2, 141– 161. Haff, W., Friedrich, K., Mayr, G., Schlaich, J., 1983. Solar chimneys, part I: principle and construction of the pilot plant in Manzanares. International Journal of Solar Energy 2, 3–20. Markowski, P.M., Straka, J.M., Rasmussen, E.N., 2003. Tornado genesis resulting from the transport of circulation by a downdraft idealized numerical simulations. American Meteorological Society, 795–823. Michaud, L.M., 1975. Proposal for the use of a controlled tornado-like vortex to capture the mechanical energy produced in the atmosphere from solar energy. Bulletin of the American Meteorological Society 56, 530–534. Michaud, L.M., 1995. Heat to work convection during upward heat convection, Part I: carnot engine method. Atmospheric Research 39, 157–178. Michaud, L.M., 1999. Vortex process for capturing mechanical energy during upward heat-convection in the atmosphere. Applied Energy 62, 241–251. Michaud, L.M., 2005. Atmospheric Vortex Engine.
Solar Energy 83 (2009) 462–476 www.elsevier.com/locate/solener
Elementary theory of stationary vortex columns for solar chimney power plants N. Ninic *, S. Nizetic 1 Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Department of Thermodynamics, Thermotechnics and Heat Engines, University of Split, R. Boskovica bb., 21000 Split, Croatia Received 12 December 2007; received in revised form 30 August 2008; accepted 14 September 2008 Available online 9 October 2008 Communicated by: Associate Editor S.A. Sherif
Abstract This paper aims to develop and make use of the availability of warm, humid air. In particular, we focus on the possibility that this availibility can be concentrated at the ground level without using a solid ‘‘chimney”. The results reveal that this concentration can be achieved via the formation of an updraft ‘‘gravitational vortex column” (GVC) situated over turbines. A simplified physical and analytical GVC model is developed in this paper. A numerical solution is given for a characteristic case, with a GVC process as a part of the cycle, similar to the Brayton cycle obtained in a gravitational field. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Solar chimney; Vortex; Availability; Numerical modelling
1. Introduction The concept of a solar chimney power plant (SC) was originally proposed by Schlaich (1968). From 1980 to 1989, a prototype plant in Manzanres was developed, constructed, and tested (Haff et al., 1983; Haff, 1984; Schlaich, 1995). A detailed literature review of this field is given in Bernardes et al. (2003). According to Padki and Sherif (1989) and Schlaich (1995), desert areas and subtropical zones are of primary interest for medium and large SC plants. Solar chimney power plants can also be used under exceptional conditions (Bilgen and Rehault, 2005; Nizetic et al., 2008). The role of a concrete chimney in an SC power plant is in the concentration of a part of the working availability (of heated air leaving a solar collector) at the chimney inlet. *
Corresponding author. Tel.: +385 21305955; fax: +385 21463877. E-mail addresses: [email protected] (N. Ninic), [email protected] (S. Nizetic). 1 Tel.: +385 21305948; fax: +385 21305954. 0038-092X/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2008.09.002
Part of the working availability used by the chimney is concentrated at the bottom of the chimney and is in the form of the mechanical non-equilibrium. The pressure in the chimney is lower than the atmospheric pressure at the same level. That form of non-equilibrium can be extracted using turbines. In this paper, the idea for the replacement of a chimney (of a limited height) with a vertical gravitational vortex column is proposed. In the optimal case, the height of the vortex column would be equal to the thickness of the troposphere layer. At the bottom of this stream structureatmospheric gravitational vortex (GVC) (Ninic, 2006), a significantly low pressure (compared to the atmospheric pressure at the collector outlet) would be achieved. The idea of using an atmospheric gravitational vortex as a flow object to play the role of a convective chimney is not a new one. It was originally presented in Dessoliers (1913). In the 1950s, Dessens performed an experiment that, according to Dessens (1962, 1969), proved that such an idea was possible in principle. The idea of using an atmospheric gravitational vortex as a specific heat engine was proposed and somewhat developed by Michaud (1999,
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Nomenclature a ek ep enet coll tech g h h+ M m_ m_ R m_ 01 m_ 12 P p Q_ coll Q_ out R s t m wt wtc wtp wc wfr wR wz 0 1 wz 1 2 wz x zc
angular momentum (m2/s) specific kinetic energy (J/kg) specific potential energy (J/kg) the net technically feasible part of exergy or height potential (J/kg) gravitational acceleration (m2/s) specific enthalpy of air (J/kg) specific total enthalpy (J/kg) momentum (Nm) mass flow rate (kg/s) radial mass flow rate (kg/s) mass flow rate through the internal GVC layer mass flow rate through the GVC updraft shell resultant turbine power (MW) pressure (Pa) heat absorbed in the collector (J) rejected heat (J) radius (m) specific entropy (kJ/kgK) temperature (°C) specific volume of air (m3/kg) specific shaft work (J/kg) specific shaft work in the central turbines (J/kg) specific shaft work in the peripheral turbines (J/ kg) circular velocity (m/s) specific internal friction work (J/kg) radial velocity (m/s) vertical velocity (m/s) vertical velocity of the internal GVC layer vertical velocity of the GVC updraft shell mixing ratio (kgd.a/kgv.) – mass of moisture per unit mass of dry air outlet height (m)
2005). Specifically, Michaud (1975) proposed a ‘‘vortex power station” with heated air, creating a vortex in the atmosphere. The previous works of Haff et al. (1983), Michaud (1995) and Ninic (2006) investigated the magnitude of the working availability of warm air in the atmosphere. In the last study, the term for the technically useful part of the working availability of collector air was worked out in detail. It was defined as the technically feasible part of the working availability in a gravitational field: work gained by reversible adiabatic expansion to a state of stable mechanical equilibrium with the surrounding atmosphere (at a maximum accessible height of zmax). The technically feasible part was defined by
enet coll tech
¼
Z 0
zmax
dpa ðzÞ mcoll ðzÞ g dz: dz
ð1Þ
zmax d Dz g gt P(z) q u
maximum height (m) thickness of the GVC downdraft shell (m) height step (m) dynamic viscosity (Pa s) heat to work efficiency characteristics of the downdraft shell (kg/m2) density of air (kg/m3) relative humidity of air
Subscripts 0 axis a atmospheric c circular (component) velocity coll collector k kinetic energy p potential energy R radial direction (inward) z component of vertical velocity 1 at radius R1(z) 2 at radius R2(z) Double 00 01 02 11 12
subscripts collector outlet state in the axis at z = 0 state in the axis at z = 10,000 state at radius R1, z = 0 state at radius R1, z = 10,000
Superscripts stagnation state without wc with wz + stagnation state without wc and with wz fr internal friction real state with internal friction 0
Here, mcoll(z) is the adiabatic function of the specific volume of air present in the collector, pa(z) is the surrounding atmospheric pressure at the same height, and g is the gravitational acceleration. The integral (1) is equivalent to the work of the buoyancy force affecting 1 kg of the collector air that has left the collector. In this paper, we refer to integral (1) simply as the air collector ‘‘height potential”. Part of enet coll tech would be used by ground level turbines of a solar chimney power plant with a significantly shorter chimney. The other part of enet coll tech would be used for maintenance of the GVC flow structure and to overcome the internal friction. To provide the optimal case, we will later specify zmax as the upper troposphere boundary: z = zmax = 10,000 m with vertical profiles of pressure and temperature in the atmosphere taken from the standard (NOAA, 1976) atmosphere. This paper proves that, to obtain enet coll tech work by means of vortex structure, heat rejection is necessary after the air
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has passed the turbine. This heat rejection can be realised in two ways: – immediately after passing the turbine, or – at the GVC top. Physical and mathematical models for the latter case are developed later in this paper. 2. Starting assumptions and boundary conditions We suppose that stationary updraft airflow exits a short chimney positioned over the turbines. The updraft has a spiralling upward flow, so that its pressure on the inside is lower than the atmospheric pressure corresponding to the same height. The sub-atmospheric pressure behind the turbines is analogous to the sub-atmospheric pressure maintained in SC-power plants by the chimney effect. The fact that the sub-atmospheric pressure difference inside a GVC would be highest just above the ground means that there is a concentration of air height potential from the collector (1) close to the ground. Hence, the only place where air turbines can be installed is near ground level. The updraft spiralling flow occurs adiabatically with maintenance of angular momentum in only one part of the GVC cross-section, around the radius R1(z) of the central zone. This can be seen in Fig. 1. The goal is to try to avoid the intense internal friction that would occur in the central
zone in the case of strict maintenance of angular momentum. Circular updraft airflow as a solid body is therefore established for R(z) < R1(z) (Fig. 1) by a particular boundary condition. In this respect, the boundary conditions for the GVC are as follows. (A) At the level of the short chimney outlet, where the independent variable z is taken as zero (z = 0), the GVC central section is generated above the central group of turbines for R1 > R > 0 and is taken to be rotating as a solid body and moving upwards with a homogenous vertical velocity w0z(z). For the part of boundary a-b in Fig. 1, we therefore have wz0 ðzÞ ¼ wz0 ð0Þ;
ð2Þ
and for the circular velocity component along the same boundary a–b: wc ð0Þ ¼ wc1 ð0Þ
R ; R1 ð0Þ
ð3Þ
where wc1 ð0Þ is the circular velocity at radius R1(0). The pressure distribution, providing a radial pressure equilibrium for velocity distribution (3), is a known function; it depends on the pressure in the vortex axis p0(0) and rises up to p1(0) at R1(0). It is formulated in Eq. (19) in Section 7. (B) The pressure of the GVC axis p0(z) for 0 < z < zmax is established as a boundary condition along the a–c boundary shown in Fig. 1. This pressure is defined by the vertical isentropic flow of the section of air originating from the
Fig. 1. Two-cell model of the axisymmetrical flow of a gravitational vortex column (with boundary conditions along the broken contour).
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collector. This section of air flow has passed through the central turbine (where pressure is p0(0)) at the short chimney outlet. The connection between p0(0) and p0(zmax) = pa(zmax) depends on the vertical velocity change w0z(z). In the simplest case, when this velocity is either negligible or constant, the function p0(z) simply reflects the hydrostatic relationship: Z zmax q0 ðzÞ g dz; ð4Þ p0 ðzÞ ¼ pa ðzmax Þ þ z
where q0(z) is an isentropic function of pressure p0(z). Generally, p0(z) is a rather complex function that is also dependent upon w0z(z). In any case, the pressure on the GVC axis at the level of the short chimney outlet must be considerably lower than the atmospheric pressure at the same height. (C) The third set of boundary conditions comprises those conditions imposed on the cross-section through which air that has left the main (peripheral) turbines enters the short chimney outlet. Here, the air moves upwards with a radially homogenous vertical velocity 1 w2z ð0Þ, maintaining angular momentum according to the ‘‘free vortex” law. This corresponds to the b–d boundary in Fig. 1, with R2(z) > R(z) > R1(z) for z = 0. We will call this flow or ‘‘layer” within the GVC, which is concentric with the central section, the ‘‘updraft shell”. This flow is established by the given angular momentum a that: wc ð0Þ Rð0Þ ¼ wc1 ð0Þ R1 ð0Þ ¼ const: ¼ a;
ð5Þ
and by the given radial homogenous vertical velocity that: 1
w2z ð0Þ
ð6Þ
for R2 ð0Þ > Rð0Þ > R1 ð0Þ: The radial pressure distribution in this GVC section is defined by (5) and p1(0), in connection with boundary condition A (Fig. 1). The pressure distribution along the b–d boundary section will be formulated in Section 7. (D) The upper GVC boundary is at the top of the troposphere for z = zmax, the c–e contour in Fig. 1; the pressure is atmospheric pa(zmax) = 0.264 bar, R1(zmax) ? 1, R2(zmax) ? 1, and all velocities disappear. The boundary conditions for the horizontal plane at the short chimney outlet, at the axis, and on the top of the troposphere are defined by A, B, C, and D. Model in Fig. 1 represents the vortex column composed by two ‘‘layers”: an updraft central flow and the updraft shell around it. The states of the air in the two layers are different, although both flows belong to the same air collector adiabat. The only remaining undefined boundary condition is that along the outside d–e contour in Fig. 1. Upon first approximation, and from a purely mechanical perspective, these boundary conditions are: pressure is equal to the atmospheric pressure p2(z) = pa(z), and velocities are wc2 ðzÞ and 1 w2z ðzÞ. Mixing and friction with the atmosphere at
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the GVC periphery are treated as negligible in this elementary model. However, the whole process cannot be based on mechanical considerations alone. It must be verified whether the assumed two-cell flow-structure is compatible with the requirements of thermodynamics. 3. Thermodynamic aspects For the vortex column’s peripheral radius R2(0) (point d in Fig. 1), with the boundary condition (5), the pressure p2(0) should be equal to the atmospheric pressure: p2 ð0Þ ¼ pa ð0Þ:
ð7Þ
If the collector air pressure at the collector outlet and the turbine inlet pcoll (point B in Fig. 1) are the same as the atmospheric pressure close to the ground (z = zc), then it holds that: pa ð0Þ ¼ pcoll qa ð0Þ g zc :
ð8Þ
Two very close isobars pa(0) and pcoll = pa(zc) are shown on an h–s diagram in Fig. 2. The state of atmospheric air close to the ground (A) and the state of atmospheric air at the short chimney outlet height (F) are plotted in Fig. 2 (G is the auxiliary state of isentropically expanded ground air up to the same height). Point B is the state of air at the collector outlet, with enthalpy equal to hB and with negligible kinetic energy. If the air did not pass through the turbine, it would have a reduced enthalpy of hC at the chimney outlet, a potential energy of gzc, and stateRC (Fig. 2). This potential energy is F equal to the integral A va dp over the vertical profile in a stable atmosphere (AF in Fig. 2) (see, for example, Ninic (2006)). The potential energy is approximately equal to the isentropic difference in enthalpies hA hG. Hence, the
Fig. 2. Air states between the collector inlet and the short chimney outlet given by the GVC two cell model.
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state at the short chimney outlet – without any work of the turbine and without any gain in kinetic energy – would be C. This state is defined by the line GC, which is equidistant from the isobar AB. The kinetic energy ek, which in this state could be obtained by expansion up to height z = 0, would be hC hE, according to the First Law of Thermodynamics. The pressure at point d in Fig. 1, at the GVC periphery, is pE. Kinetic energy hC hE is gained by the assumption that no work is delivered to the turbine and that the elevation and acceleration process has no internal friction. State D would be the outgoing state of the process with internal friction during elevation/acceleration. Let us evaluate the order of magnitude of this kinetic energy at state E, ek(E), for the short chimney height, zc = 10 m, and for a temperature increase in the collector from 15.0 °C in state A to 45.0 °C in state B. Approximately, this gives hB hC ffi hA hG ffi va ðpcoll pa ð0ÞÞ ffi gzc ¼ 98:1 J=kg hence; pcoll pa ð0Þ ffi 98:1=0:82 ffi 119:0 Pa hB hE ffi vB 119 ffi 0:91 119 ffi 108:0 J=kg hC hE ffi 108:0 98:1 ffi 9:9 ffi ek ðEÞ J=kg; hence the velocity at state E is wðEÞ ffi 4:44 m=s: In the case of zero kinetic energy at the turbine outlet, the relatively small corresponding kinetic energy (9.9 J/ kg) could be replaced by a theoretical amount of work in the turbine. In that case, the total enthalpy after the turbine at radii R2(0) is hE (Fig. 2), and the total work performed in the turbine is hB gzc hE = 9.9 J/kg. Before we make any conclusions based on the previous result, we should discuss an additional issue: can the air at the smaller radii R(0) (i.e., for R2(0) > R(0) > R1(0)) have a total enthalpy lower than hE and provide work greater than 9.9 J/kg? According to the free vortex radial distribution (5) established in the above-mentioned range of radii, it holds that 2 w ð9Þ mdp ¼ d c 2 ; 2
where h0 ð0Þ ¼ hð0Þ þ w2c ð0Þ=2 ¼ const:
ð11Þ
The part of the stagnation enthalpy due to vertical velocity is omitted because, according to (6), this component is constant. Therefore, the stagnation enthalpy h0 does not depend on the radius of the short chimney outlet. This means that all parts of the updraft shell in the radii interval R2(z) > R(z) > R1(z) must deliver the same amount of work to the turbine. Thus, we can conclude that the maximum possible work delivered by the turbine is 9.9 J/kg. Comparing this result for work done in the turbine with the ground-level effect and air velocities in natural tornados, one may conclude that the supposed two-cell GVC model can neither be the basis for a realistically functional GVC device, nor can it form the basis for explaining the destructive power of natural tornados. Now, let us analyse another possible case. Let the pressure behind the turbine at radius R2 be considerably lower than the atmospheric pressure at the same height (p2(0) < pa(0)). Such a state corresponds to point H in Fig. 3, instead of points E in Figs. 2 and 3. In Fig. 3, the potential energy increase up to the chimney outlet is gzc, the shaft work thus obtained in the turbine is wt, and the kinetic energy behind the turbine is ek. This setup allows for the production of considerably more work and kinetic energy, but with one condition: the air passing through the turbine must be able to attain atmospheric pressure at the R2(0) radius (state J in Fig. 3). Before that, this air must reject a certain quantity of heat, Dq in Fig. 3. The reason for this is the same as in the case of condensate return in the steam boiler after heat rejection occurs in the condenser. More sophistically, this is due to the divergence of isobars in the h–s diagram, as a consequence of the Second Law of Thermodynamics. State J could be achieved by
with 1 w2z ð0Þ ¼ const. Relation (9) means that all air states in this cross section belong to the same isentrope (passing through B in Fig. 2). Therefore, according to the First Law of Thermodynamics, in this range of radii we have mdp ¼ dh:
ð10Þ
Hence, from (9) and (10), it follows that the specific stagnation enthalpy h0 is constant over the same cross section: dh þ dðw2c =2Þ ¼ dh0 ¼ 0; 2
See Section 7.
Fig. 3. The states of air between the collector outlet B and the short chimney outlet H.
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the transformation of kinetic energy (ek(I)(=ek(H)) into pressure increase p(J) p(I)). Based on this discussion and on the fact that in nature there exists quasi-stationary tornado funnels with very destructive effects at ground level, we can conclude that the existence of a GVC column with a near ground concentration of availability (for turbine work) is possible, because of the fact that it exists in nature. However, it cannot be modelled by a two–cell vortex without heat rejection. In contrast, according to Fig. 3, with heat rejection it should be p2 ð0Þ ¼ pðHÞ < pa ð0Þ:
ð12Þ
Therefore, a considerable supplementation of the two-cell model is necessary in order to bridge the pressure ‘‘gap”, pa(0) p2(0). There are only two possible ways of achieving this, one of which is proposed and developed here. 4. Physical model formulation According to the previous section, after adiabatic expansion in the peripheral turbine (in Fig. 1), the collector
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air can only regain its high initial pressure (atmospheric pressure at close to ground level) if it previously rejects a sufficient quantity of heat. The first way to achieve this is by filling the zone around radius R2 with high pressure airflow (pa > p > p2). This airflow is provided from air that has passed the peripheral turbines and reached the top of the GVC, where it rejects the heat. After heat rejection, cooled air descends toward the ground, rotating around the updraft flow at radii R2(z) (flow E in Fig. 4). We can suppose that this descending airflow would have the form of a rotating annular shell of thickness d(z), which would embrace the updraft shell. This physical model is named ‘‘three-layer” model. On the other hand, cooled air descending from height zmax in the standard stable atmosphere could not descend without containing a large amount of condensate or ice droplets in its volume. It would be a downdraft induced by falling precipitation. Adiabatic saturation with steam evaporated from condensate droplets increases the cooled air density and contributes to its descent. Condensate droplets in the descending shell arrive in the descending shell via
Fig. 4. A simplified three-layer physical model of the GVC stationary vortex column.
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the updraft shell. The descending shell, due to mixing with the atmosphere, has a specific angular momentum lower than the initial one (a0 6 a). The external shell movement can therefore be assumed to be circular and downward (see Fig. 4), yet, overall, it is considerably more complex. 4.1. Second mechanism for pressure gap over-bridging There is another way of rejecting heat from the air after it passes through the turbines and regains atmospheric pressure. This consists of bringing the air at the turbine outlet into contact with cold, low-pressure external air and allowing the two to mix. This can be achieved via a weakly forced central downdraft of cold, high-level atmospheric air. These air parcels would stop and mix with collector air over the turbine, thus allowing the collector air to obtain atmospheric pressure (Ninic and Nizetic, 2007). This model of a GVC is named the ‘‘central downdraft model”. Some sources, for example Bluestein (2005), have indicated that such contact with high, cold air occurs at a certain low altitude above the ground in the central zone of natural tornados and ‘‘dust devils”. The effect of cooling by the central descending airflow could be increased and/or initiated by injecting liquid water into the space behind the turbine. 4.2. Additional validation of the physical model with a downdraft annular shell In addition to the thermodynamic validation of the three-layer model (Fig. 4), we can also check the idea by using existing numerical models of quasi-stationary tornadoes, such as those in the work of Markowski et al. (2003). In this model, which is inspired by observations and experimental readings made in the field with natural tornadoes, the authors concluded that a descending airflow (in the form of a rotating shell) is necessary for tornado development to quasi-stationary states. In this paper, as in our three-layer model, the radial pressure drop corresponds to the centripetal acceleration in the rotating downdraft shell. We also note that, in consideration of tornado genesis, the emphasis is on the important role of the descending and rotating annular shell, as well as on the transmission of angular momentum a0 . This angular momentum is initially at its maximum at the top of the rotating annular shell and at approximately zero near ground level. However, our proposed three-layer GVC model does not have solely tornado-specific characteristics, and the genesis of GVC is not analogous to that of tornadoes. In Bluestein (2005) and Snow and Randal (1984), tornadoes and dust devils were classified as ‘‘one celled” or ‘‘two celled.” The ‘‘two celled” types are in good agreement with the GVC ‘‘central downdraft model”. Our analysis provides conclusions that are in agreement with the observations of natural vortex structures, including the bridging
Table 1 An explanation of the labels used in Fig. 4 Label
Label description
A T1 T2
Collector roof Annular space for peripheral turbines with electric generators Turbines defining flow B in the GVC central part, rotating as a solid Flow of ascending vortex air shell leaving the peripheral turbines Heat sink, i.e. the space in which collector air delivers heat to the environment ‘‘Black box” – descending spiralling shell
C D E
of the pressure gap in the ‘‘three-layer model” (with a downdraft outer shell). Furthermore, we developed the simplified the threelayer GVC model based on the following propositions: – Pressure ‘‘gap” filling is achieved by the externally rotating descending shell. – For the GVC to reach the top of the troposphere, the appropriate radial pressure equilibrium condition for a rotating descending shell (thickness, effective density, and circular velocity) should be satisfied. The descending shell structure could be composed of a mixture of saturated air and liquid or ice droplets. The complex structure of the shell and its internal motion is not described in the proposed three-layer physical model. Descending shell volume is a boundary condition and is in the form of a ‘‘black box” that serves to fill the pressure gap between the GVC upflow and the atmosphere. Vertical and cross section views of such a GVC model are shown in Fig. 4, with explanation of the labels given in Table 1. Fig. 5 is a schematic of the vertical section of one-half of a GVC, with boundary conditions underlined. These conditions are defined at the level of the short chimney outlet (z = 0), along the axis and at the top level of the troposphere (z = 10,000 m). Side contact with the atmosphere is shown as a band and not a line. This band is a descending rotating shell with an unknown internal structure. This structure is characterized by the function P(z), which determines a mechanical radial equilibrium with the atmosphere. This function is analytically defined in Section 7. 5. Internal friction and dissipation of angular momentum Fig. 4 shows profiles of three axially symmetric layers that are characterized by their pressure (p), circular velocity (wc), and vertical velocity (wz) distributions in the cross section of the simplified GVC model. The rising GVC flow, 0 < R(z) < R1(z), rotates as a solid body (a ‘‘forced vortex” in Rankine’s terminology). According to Rankine’s ‘‘free vortex law”, the annular rising flow, (R1(z) < R(z) < R2(z)), is the section of spiral air movement from the peripheral turbines. The third layer belongs to the descending condensate and cooled air shell, which also has a circular velocity component. The linear distribution of circular velocities in
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of radius R1. The stationary free vortex flow is such that the moment of the friction force, by which each fluid layer affects the adjacent one, is independent of the radius. The viscous force moment per unit height of the vortex equals (Binder, 1958): M ¼ 4pgwc R:
ð13Þ
Due to the independence of this expression from the radius, it follows that wc R ¼ const: ¼ a; which is the ‘‘free vortex law”, as in our model updraft shell. The same angular momentum M would be lost per second at the outer cylindrical surface (R = R2) of such a flow with viscous force momentum. Suppose that this dissipation of angular momentum is compensated by a centripetal radial flow without angular momentum. The flow per unit height is m_ R . This yields 4pgwc R ¼ m_ R wc R ¼ qwR 2pRwc R; where wR ¼ Fig. 5. Boundary conditions of the GVC section.
the central rotating flow is not established spontaneously by friction, but rather is imposed at the outlets from the central turbines. The primary source of internal friction in annular rising flow is associated with differences in vertical velocities at the radius R2(z). This includes the secondary effect of ‘‘heat addition”, analogous to the ‘‘heat return” effect found in multistage turbine processes. Beyond the loss of availability, friction on boundary surfaces may also result in an outward radial transfer, as well as a radial loss of angular momentum in the direction of a static atmosphere. The neglect of the radial dissipation of angular momentum can be justified by one of the following reasons (or by a combination of them). The first possible reason is a low intensity of radial angular momentum loss per square meter in the rotating flow. The second reason for the relatively small angular momentum dissipation could be the small ratio between the perimeter and the cross sectional area of the corresponding GVC flow section. Finally, a third reason for such a small dissipation could be the existence of at least a small radial air inflow, which is sufficient to compensate for most of the radial angular momentum dissipation by friction. For a rough numerical estimate of these arguments, let us first examine the entire laminar flow of a viscous fluid extended to infinity and moving around a rotating cylinder
2g qR
ð14Þ
is the centripetal radial velocity of the surrounding air by which the loss of angular momentum in the environment would be fully compensated. For a numerical estimate of wR, we anticipate the dimensions and velocities in one characteristic GVC with R2 = 15 m and wc2 ¼ 60 m=s for g = 106 Pa s. Hence, from (13) and (14), M = 0.0113 N/m, m_ R R2 wc2 ¼ 14; 130; 000 kg m=s2 , and wR = 0.00000013 m/s. Therefore, in this case, the radial inflow necessary to compensate for the momentum radial dissipation is negligible in relation to the corresponding overall angular momentum flow. The phenomenon of instabilities in annular rotating upflows enveloped by rotating downdrafts has not been well studied in the literature; this is a subject in need of additional research effort. Because of this, we can only rely on the previous works of Taylor and Prandtl, elaborated in Schlichting (1960). In these works, the limits of stability were defined in the case of laminar flow, but for circumstances different than those in our case. Their work is related to an annular rotating fluid flow which is set between a rotating cylinder of radius R1 and a cylinder at rest (which has an appreciably greater radius). The situation here, with the annular space of the rotating fluid lying between the sections of the updraft annular flow and the downdraft annular flow (together from Rmin = R1 to Rmax = R2 + d) is relatively well described by such circumstances. According to Schlichting (1960), we can conclude that instability appears according to the following criterion:
470
wc1 ðR2 R1 þ dÞ m
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2 R1 þ d > 41:3: R1
ð15Þ
This instability is caused by a centrifugal force that is higher on the fluid particles near the inner radius R1. Instability results in the appearance of cellular vortices with axes directed along the circumference. Anticipating the dimensions and velocities in the same numerical example, we can conclude that the dimensionless left side of the inequality criterion is rffiffiffiffiffi 60 10 10 ffi 36 1015 : 15 1015 12 This is far beyond the stability limit value, and it even stretches considerably into the domain of turbulent flow. Closed streamlines of secondary motions (caused by instability), however, are fully inside the space of free vortex flow and have a homogeneous radial distribution of angular momentum. In this case, intensive internal friction in the updraft’s annular flow does exist, but the radial flow of angular momentum is not very significant. These conclusions are supported by some well-known observations made on natural tornado ‘‘funnels” that have been published in the Encyclopaedia of Science Technology (1978). The first such observation is that quasi-stationary tornadoes may exist for several hours without a renewable source of angular momentum. The second observation regards the way in which a tornado funnel disappears; it does not disperse radially in the atmosphere, but rather becomes more slender, until it takes the form of a rope that eventually breaks. Overall, this supports the hypothesis that there is little outward loss of angular momentum through the surface R2(z). Furthermore, we may conclude that the downdraft shell (with R(z) > R2(z)) has a structure that is more complex than that of the updraft shell. In this simplified model, the downdraft shell is treated as a ‘‘black box” that serves to fill the pressure gap: pa(z) > p(z) > p2(z). In neglecting the radial loss of angular momentum, we do not neglect the internal friction caused by the viscosity of the fluid and by different forms of motional instability. However, in this simplified analysis, the internal friction Dwfr in the energy equation written for the stream line along the annular updraft flow3,
In conclusion, we can say that, in this paper, the internal friction is still taken into account, but the unused part of availability at the top of the GVC has been purposely left out. The influence of internal friction can be investigated by varying the unused quantity of availability at the top of the GVC. 6. Additional boundary conditions In this section, we complete the boundary conditions defined in Section 2. Let us start from the section describing a short chimney outlet on the ground level, z = 0. The radius R1(0) is chosen to be of sufficient magnitude such that circular movement of air with constant angular momentum is provided for R(0) > R1(0). For R2(0) > R(0) > R1(0), relation (5) is satisfied. Here, the pressure distribution along the radius at z = 0 is such that radial equilibrium is fulfilled. Accordingly, the pressure gradient is proportional to the centripetal acceleration of circling fluid particles: dp ¼ qw2c
dR ; R
ð16Þ
with (5): w3c adwc 2 ¼ wc dwc ; mdp ¼ a wc vdp ¼ wc dwc ; which is anticipated by the previously stated relation (9). According to the First Law of Thermodynamics for reversible fluid flow: dq ¼ dh vdp:
ð17Þ
The imagined reversible adiabatic movement of a fluid parcel along the cross section in updraft shell is obtained by dh ¼ mdp ¼ wc dwc ; hence, h þ w2c =2 ¼ h0 ¼ const:;
ð18Þ
0
where h is the corresponding stagnation enthalpy. The vertical component of the velocity 1 w2z ð0Þ is homogenous over the cross section at a given radius and contributes to the total stagnation enthalpy h+:
vDp ¼ Dep þ Dek þ Dwfr ;
hþ ¼ h þ ðw2c þ w2z Þ=2 ¼ const:;
does not need to be expressed explicitly (as Dep and Dek). The process along the streamline will be treated by neglecting internal friction (without Dwfr4), but with a requirement that a certain positive value of availability in the annular rising flow at the top of the GVC is retained. By accomplishing this, in this simplified model we have ‘‘reserved” a place for Dwfr in the simplified energy Eq. (21).
as a property independent of radius R, for R1(0) < R(0) < R2(0). Based on the previous equations, and according to Fig. 5, the completed boundary conditions are the following. (A) At the level of all turbine outlets, the radius R1(0) and vertical velocities wz(0) for both the central and the peripheral turbines (more precisely for the short chimney outlet, at), are given in Figs. 4 and 5. The pressure gap pa(0) p2(0) is also a feature of the boundary conditions connected with downdraft shell. Pressure, temperature,
3 4
See Eq. (210 ). See Eq. (21).
ð180 Þ
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mixing ratio x, and mass flow rates are all given at the collector outlet. The mass flow rates are: m_ 01 through section 0–1, m_ 12 through section 1–2, and the overall mass flow rate is m_ ¼ m_ 01 þ m_ 12 . These data also define the profiles of all air properties at height z = 0. (B) Temperature, pressure, and density profiles along the GVC axis are imposed by the vertical airflow from the central axial turbine. This means that the pressure on the axis at 10,000 m is the atmospheric pressure at that height, and that other properties along the axis are determined by the reversible adiabat of the upward moving collector air. For the simplest case, the distribution of pressure p0(z) corresponding to constant vertical velocity wz0 is shown in Fig. 6. This shows the stepwise rise of pressure near the top of the troposphere due to the transformation of kinetic energy from ðwz0 ðzmax ÞÞ2 =2 to Dp/q (as explained in Section 2, paragraph B). In principle, taking into account the variation of wZ 0 ðzÞ with height does not cause complications, except for the additional imposition of continuity and the First Law of Thermodynamics for axial flow. In any case, whether constant or variable, the velocity wz0 ðzÞ plays a double role: 1. To ensure the presence of the central turbine air flow in the axis of GVC. 2. To ‘‘translate” the static distribution of pressure p0(z) 2 towards lower values for q½wz0 ðzÞ =2. For an illustration of this, compare the full line p0(z) and the broken line shown in Fig. 6. (C) There is a (standard) atmospheric pressure profile on the outer side of the downdraft shell. In this model, the effect of the downdraft shell on the GVC updraft shell is only reduced to radial pressure difference. (D) At height zmax = 10,000 m, the environmental temperature is 50.0 °C, the pressure is 0.264 bar, and practi-
cally all moisture is in the form of small ice crystals. We assume that the air forming the downdraft shell has an initial pressure and temperature equal to the corresponding atmospheric pressure and temperature, containing additionally all the moisture in the form of condensate. As such, the downdraft descends adiabatically, including a constant saturation process due to the presence of the condensate. Saturation by vapour and the presence of a sufficient quantity of condensate are prerequisites for this air to descend in a stable atmosphere. 7. Analytical formulation In the analytical articulation of the physical model presented here, in proceedings of paper we will try to ensure that the relations used are not of speculative nature, with potential applications only in particular circumstances. Rather, we will try to ensure that they are based on general principles of mass, momentum, and conservation of angular momentum, the First Law of Thermodynamics for various observers (from various frames of reference), and the Second Law of Thermodynamics. Additionally, the equations used to describe humid air states in a psychrometric chart will be assumed to be applied throughout. First, we summarize the unknown functions of height (z), on which any geometrical or physical properties of any GVC cross section may depend. With the given boundary conditions, such properties are fully defined in terms of five unknown functions of height: R1(z), p1(z), 1 w2z ðzÞ, R2(z), and p2(z). The radial pressure distribution and circular velocities are defined by R1(z), p1(z), a and p0(z), within the interval 0 < R(z) < R1(z). The vertical velocity in this interval is homogenous, approximately constant with regard to height, and equal to 0 w1z ð0Þ. The temperature and density distributions within the same interval 0 < R(z) < R1(z) are also defined by the fact that all states of air originating from the collector belong to the same isentrope. The distribution of all properties within the interval R1(z) < R(z) < R2(z) is defined by R2(z) and p2(z), including circular velocities, while the vertical velocity is homogenous and equal to 1 w2z ðzÞ. The five listed unknown functions at height z + Dz are derived from their values at the previous level z, from five independent equations, and from the side boundary conditions. All five equations are conservation laws of general significance. The first equation is the radial pressure equilibrium condition (17) applied within the radius interval 0 < R(z) < R1(z) to fluid rotating as a solid body (according to law (3)). By integration, we get p1 ðzÞ p0 ðzÞ ¼
Fig. 6. Illustration of axial pressure distribution along an axis and the effect of wz0 disappearing near the top of the troposphere.
471
10 1 q ðzÞw2c1 ðzÞ; 2
ð19Þ
where 0 q1 ðzÞ is the corresponding average density of the fluid. The second relation is the conservation of the mass flow rate for the updraft shell:
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df1 w2 ðzÞ1 q2 ðzÞ½R22 ðzÞ R21 ðzÞg ¼ 0:
ð20Þ
The third relation is derived by applying the First Law of Thermodynamics from the view of two different thermodynamic observers. In its general form, it is5: v1 ðzÞdp1 ðzÞ ¼ dep1 þ dek1 þ dwfr ¼ gdz þ w1 ðzÞdw1 ðzÞ þ dwfr :
ð210 Þ
Here, w1(z) is a module of the velocity vector at radius R1 ðzÞðw21 ¼ w2c1 þ w2z Þ, and dwfr(P0) is the work due to internal friction. Neglecting this friction work for the time being, we get m1 ðzÞdp1 ðzÞ ¼ gdz þ 1 w21 ðzÞdw1 ðzÞ:
ð21Þ
The motivation for neglecting friction work was explained at the end of Section 5. The fourth independent relation is the radial equilibrium condition for the fluid inside the updraft shell (17); integrating (17) with (16) gives p2 ðzÞ p1 ðzÞ ¼
11 2 q ðzÞ½w2c1 ðzÞ w2c2 ðzÞ: 2
ð22Þ
Finally, the radial pressure equilibrium condition for the downdraft shell (modelled as a”black box”), also according to (17), is pa ðzÞ p2 ðzÞ ¼ ðqdÞðzÞ
w2c ; RðzÞ
ð23Þ
where qd(z) is the product qd as a function of height. Here, pa(z) is the atmospheric pressure at height z; q and d are the effective density and downdraft shell thickness, wc ðzÞ is the effective circular velocity, and RðzÞ is the average radius of this shell. Eq. (23) can be specified by pa ðzÞ p2 ðzÞ ¼ PðzÞ
w2c2 ; R2 ðzÞ
ð24Þ
where PðzÞ ¼ ðqdÞ
w2c w2c2
R2 ðzÞ : RðzÞ
upon which the whole GVC structure depends. This includes its state at the short ‘‘chimney” outlet section. In the following sections, we present a detailed numerical solution for a particular GVC chosen as characteristic of many solutions with different adopted constructive parameters, plant parameters, and corresponding height stratification of the values of the P(z) functions. The adopted atmosphere was standard in all cases (NOAA, 1976), its maximum height being 10,000 m. Atmospheric temperature, pressure, and density at the ground level are 15.0 °C, 1.0132 bar, and 1.225 kg/m3; at the height of 10,000 m, temperature, pressure, and density are 50.0 °C, 0.26436 bar, and 0.413 kg/m3, respectively. Ideal gas relations are used in the thermodynamic calculations of moist air properties. According to ASHRAE (2005) (section: Thermodynamic properties of moist air), errors in enthalpy and specific volume calculations at standard atmospheric pressure for a temperature range of 50 °C to 50 °C are less than 0.7%. These errors decrease with a decrease in pressure. 8. An algorithm and characteristic numerical solution For the numerical calculations, our algorithm starts from the first numerical step of the Dz increment and continues with the assumed, relatively low vertical velocity of the central flow (this is assumed on the axis as well). The pressure effect of a vanishing low vertical velocity is only added in during the last step of the algorithm. Vanishing of the velocity is connected by the abrupt extension of the radius R1(z) immediately before z = 10,000 m. The initial parameters and boundary conditions are shown in Table 2 (all states of air in Table 2 are marked according to the notation in Fig. 7). Qualitatively, this and other states are shown in the p–m diagram in Fig. 7. Values for Table 2 Initial parameters and boundary conditions
ð25Þ
The P(z)-function functions in boundary condition C, Section 6, which specifies the downdraft shell as a ‘‘black box”. The downdraft shell is separated from the rest of our simplified model, as an independent boundary condition. The relations established by (19), (20), (21), (22) and (24) compose a system of five equations with five listed unknown functions of height. The function P(z) in Eq. (24) is not an unknown function but a boundary condition 5 We note in passing that the relation (210 ) can be obtained from the rate of change of momentum multiplied by the fluid parcel vertical displacement. Thermodynamic derivation (210 ), emphasized here, is more general and based on the First Law of Thermodynamics for two different observers: one is immobile with respect to the parcel in motion, while the other is immobile with respect to the Earth’s surface. The connection between the mechanical and thermodynamic aspects of Eq. (210 ) is discussed in more detail in Ninic (2008).
Ground-level state of standard atmosphere, denoted by ‘‘1a” t1a(0) = 15.0 °C; pa1(0) = 1.012 bar; q1a(0) = 1.225 kg/m3; pa(10,000) = 0.26436 bar h1a = 15,075 J/kg (enthalpy of dry air) Air state at the collector outlet denoted by ‘‘00” t00(0) = 45.0 °C; p00(0) = 1.012 bar; x00(0) = 0.0182 kgd./kgd.a.; h00 = 92,192 J/kgd.a.; s00 = 320.4 J/kgd.a. K Air state at the central turbine outlet (in the axis od short chimney), denoted by ‘‘01” t01(0) = 34.9 °C; p01(0) = 0.90818 bar; wz01 = 10.0 m/s; h01 = 82,351 J/ kg; R1(0) = 10.2 m 2 hþ 01 ¼ 82; 351 J=kg; wc1 = 58.8 m/s (a = 600 m /s). Air state at the peripheral turbine outlet in R1(0) denoted by ‘‘11” t11(0) = 36.7 °C; p11(0) = 0.92592 bar; wz11 = 30.0 m/s; h11 = 84,017 J/ kg; hþ 01 ¼ 86; 197J=kg Downdraft shell as the ‘‘black box” boundary condition P(z) P(0) = 45.7 kg/m2
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473
wt0 ¼ h00 hþ 01 ¼ 92; 192 84; 274:3 ¼ 7917:7 J=kg; where the average enthalpy is defined by þ hþ ¼ ðh þ h Þ=2:0 ¼ 84; 274:3 J=kg. The specific shaft 01 01 01 work in the peripheral turbines is wt1 ¼ h00 hþ 11 ¼ 92; 192 86; 197:6 ¼ 5994:4 J=kg:
Fig. 7. p–v diagram of atmospheric states and processes in the GVC.
enthalpy and entropy of moist air are declared to be zero according to ASHRAE (2005) for dry air and liquid water at p = 1.012 bar, t = 0.0 °C. In the presented numerical example, properties of the air state at the turbine inlet (00) (Fig. 7) are: p00 = 1.012 bar, h00 = 92,192 J/kg, s00 = 320.4 J/kgK, and t00 = 45.0 °C. The shaft work done in the central turbine is
The state 01 also dictates the whole boundary condition along the axis: p0(10,000) = 0.264 bar = pa(10,000). Other boundary conditions in this numerical example are R2(0) = 12.73 m, wc2 ð0Þ ¼ 47:1 m=s, p2(0) = 0.93236 bar, and mass flow rates m_ 01 ¼ 3314:6 kg=s and m_ 12 ¼ 5685:4 kg=s. The results are reviewed in Table 3, which gives the values of the main parameters of the process, including the heat received in the collector, the heat to work efficiency, and total turbine power. In this numerical example, the function P(z) was chosen as a boundary condition because it is compatible with the constant vertical velocity component in the central flow 0 w1z and with the mass conservation law. Additionally, the zero height in the numerical program, and in Table 3 and Figs. 8 and 9, refers to ground level, and not to the short chimney outlet height. The acquired GVC geometry is illustrated in Fig. 8. In Fig. 9, a more detailed GVC section at the height of 4000 m is shown, along with radial distributions of characteristic properties.
Table 3 State values by height Boundary conditions (m) a (m2/s) Rstart 1
w0z 1 (m/s)
w12 zp (m/s)
m (kg/s)
P[0] (kg/m2)
10.20
10.00
30.00
9000.0
45.71
Turbines and down-stream parameters h01 (J/kg) hþ h00 (J/kg) 01 (J/kg)
h11 (J/kg)
hþ 11 (J/kg)
h02 (J/kg)
hþ 12 (J/kg)
92192.0
82301.0
82351.0
84017.5
86197 6
15326.7
11804.2
wtc (J/kg)
wtp (J/kg)
Qd (MW)
gt (%)
P(MW)
7917.7
5994.42
694.053
8.7
60.325
Simulation results z (m) R1 (m)
R2 (m)
p1 (bar)
p2 (bar)
pa (bar)
w01 z (m/s)
w12 z (m/s)
wc1 (m/s)
wc2 (m/s)
P (kg/m2)
10.0 100.0 300.0 600.0 900.0 1000.0 1500.0 2000.0 3000.0 4000.0 5000.0 6000.0 7000.0 8000.0 8000.0 10000.0
12.73 12.66 12.68 12.74 13.28 13.29 13.37 13.58 14.09 14.69 15.33 16.02 16.75 17.53 18.34 Inf.
0.92592 0.91637 0.89619 0.86655 0.83767 0.82821 0.78177 0.73744 0.65559 0.58133 0.51419 0.45365 0.39914 0.35010 0.30602 0.26441
0.93236 0.92216 0.90165 0.87156 0.84301 0.83339 0.78594 0.74085 0.65826 0.58342 0.51582 0.45491 0.39998 0.35085 0.30659 0.26441
1.01205 1.00129 0.97773 0.94322 0.90970 0.89875 0.84556 0.79495 0.70108 0.61640 0.54020 0.47181 0.41061 0.35600 0.30742 0.26441
10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10 00 10.00 00.00
30.00 32.82 34.13 35.91 30.19 30.78 35.45 39.27 42.29 45.09 48.04 51.29 54.88 58.79 63.00 00.00
58.82 58.02 57.54 56.82 56.08 55.83 54.05 52.22 49.80 47.41 45.08 42.83 40.67 38.60 36.64 00.00
47.11 47.36 47.28 47.08 46.16 45.12 44 86 44 17 42.55 40.84 39.12 37.43 35.80 34.22 32.69 00.00
45.71 44.70 43.10 41.20 43.40 42.70 39.60 37.61 33.31 29.01 24 41 19.21 13.81 7.61 1.31 00.00
6000
10.20 10.33 10.42 10.55 10.69 10 74 11.10 11.48 12.04 12.65 13.30 14.00 14.75 15.54 16.37 Inf.
GVC simulator design by Sandro Nizetic.
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Fig. 8. A characteristic GVC cross section.
9. Discussion of numerical results The characteristic air states of a standard atmosphere at various heights are illustrated in Fig. 7 (curve 1a 2a). The collector air state at the collector outlet is denoted by 00, while the state at the peripheral turbine outlet is 11 (in radius R1 at the short ‘‘chimney” outlet). The state at the central turbine outlet is 01, and the state at the top of the troposphere is marked as 02 for the central turbine and 12real for the peripheral turbines. Stagnation states corresponding to the particular states are marked by a superscript +. The total enthalpy difference between the air from the central and peripheral turbines at the height of 10,000 m, þ hreal 12 h02 ¼ h12 h02 (shown in the p–v diagram), is the work capacity conserved during the isentropic process of the air passing through the peripheral turbines (particularly with R = R1(z)). This travels all the way to the top of the troposphere. At a height of 10,000 m, it takes the form of kinetic energy hþ 12 h02 . In reality, this work capacity would be consumed by internal friction during the elevation process in the updraft annular flow, according to the last passage in Section 5. However, according to that section, a distributed internal friction process is substituted for throttling 12+–12real at the top of GVC (see Fig. 7). The broken curve 00–12real represents the real process, with regularly distributed internal friction. The real state of the air from peripheral turbines is at the top of the troposphere 12real, defined by the equality of enthalpies in states 12+ and 12real.
Fig. 9. A characteristic GVC axial section, z = 4000 m.
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The air that has passed through the central turbine has obtained state 02 at z = zmax, having lost its low velocity. In this way, the mean state of the air at the top of the GVC is the result of an adiabatic and isobaric mixing of the flows of states 02 and 12real with the corresponding mass flow rates. After heat rejection at z = zmax, the difference between the air flows from different turbines disappears (therefore, there is no more need for double indices). In Fig. 7, the elements of energy transformation are thus represented in general form. The area 00p –1a –2a –12real p represents the potential energy at the top, equal to Dep = gzmax = 98,100 J/kg. Curve 00–02 shows the reversible process of the collector air starting with state 00, going to the state it would occupy at the top of the troposphere after passing through the central turbine. The area 1a–00–02– 2a, therefore, represents the height potential of the collector air as defined by (1). The state behind the peripheral turbine (or turbines) in radius R1(0) is marked by 11, and the air stagnation state is marked by 11+. According to this, the þ area 11þ p –11 –11–11p is the kinetic energy in state 11, and the area 00–11þ –11þ p –00p is the specific work done in the peripheral turbine. The process is analogous for the central turbine, with the state behind it being 01, i.e., 01+. As previously stated, the state of the updraft shell air at the top of the troposphere with internal friction is 12real. The internal friction work wfr1 would, in such a case, be þ real –12real equal to the area 12þ p –12 –12 p , and the work lost þ real due to friction is equal to the area 12þ p –12 –02–12p . As known from detailed analyses of flow processes, the internal friction work has a double effect: work loss Dwfr and heat gain are the same amount. This ‘‘heat”, however, has a secondary favourable influence on the work gained, so that the loss of work due to friction is somewhat smaller than the work of friction itself. We will use the h–s diagram to enter the actual data of characteristic states and for a presentation of the process (see Fig. 10). 00–11+ is the stagnation state change in the peripheral turbines at radius R1(0). The process performed by the collector air that has passed through the central turbine on the GVC axis is 00–01(ffi01+); the state of the air that has left the central turbine is described by pressure p01 = 0.90818 bar, enthalpy h01 = 82,301 J/kgd.a., temperature t01 = 34.9 °C, and kinetic energy ek01 = 50 J/kgd.a. In state 02 at the top of the troposphere, the pressure is p02 = 0.26436 bar (=pa(10,000)), h02 = 15,327 J/ kgd.a., and t02 = 17.4 °C. The state behind the peripheral turbines, at radii R1, is described by stagnant properties þ and tþ pþ 12 ¼ 0:27789 bar; h12 ¼ 11; 804:2 J=kgd:a: , 12 ¼ 15:3 C. The rise in entropy due to internal friction conis Dsfr ¼ s12real sþ centrated at zmax 12 ¼ 13:8 J= real kgd:a: K; s12 ¼ 320:4 þ 13:8 ¼ 334:2 J=kgd:a: K. The work lost due to friction is h12þ h02 ¼ 3523 J=kgd:a: . After heat rejection and complete equilibrium with the environment is attained, the air state is 3, as shown in Fig. 10. The air here is saturated at t3 = 50.0 °C, with negligible steam content (ffi0.1 g/kgd.a.) and with all 18.2 g/kgd.a. of water content in the form of ice. The prop-
475
Fig. 10. Mollier h–s diagram of the process in GVC and the Brayton cycle.
erties are h3 = 57,941.5 J/kgd.a. and s3 = 149.8 J/kgd.a. K. Lowered isentropically to ground level, the stagnation enthalpy of this air would be increased by the amount of potential energy (98,100 J/kg) and would, therefore, be equal to 40,158.5 J/kg. At an atmospheric pressure of 1.012 bar, saturated air at 14.5 °C has an enthalpy approximately equal to 40,160 J/kg. Its gas phase remains at about 8.0 g/kgd.a., with all liquid water in droplet form. Overall, we reach state 4, as shown in Fig. 10, with h4 = 40,158.5 J/kg. There, atmospheric air has an enthalpy of only about 15,075 J/kgd.a at ground level. The moist air in the GVC actually performs the part 00– 12real (02)–3 of the Brayton cycle in the gravitational field (Fig. 10). Internal friction in the updraft shell in process 00–12real decreases the specific work of the peripheral turbine and therefore has an important role in the structure of the GVC. A realistic cycle is much more complex than the Brayton cycle in Fig. 10. The real process of air down-drafting (after the heat rejection at a level of zmax ffi 10,000 m) is different than the one presented as the broken curves 3–4. The real state of this air after sinking to ground level is that of standard atmosphere at the collector inlet (regardless of whether all the air is used in the downdraft shell or not). The real process that leads to the standard atmospheric state at the inlet of collector is not of a local, but of a regional or even global kind. Namely, the air of state 3 (Fig. 10) is thrown into the atmosphere (as a huge reactor) on the level zmax. From the other side, the air is taken to be at
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atmospheric state at ground level. The overall heat to work efficiency is, for this reason, defined as the ratio between the work extracted from the turbines and the enthalpy difference in the collector (coming as added heat). A numerical example leads to a resultant turbine power of approximately: P ¼ 3315 7918 þ 5685 5994 ffi 60; 324:57 W ffi 60 MW: The heat to work efficiency with a standard atmospheric state at the collector inlet is gt ¼ ½P =ð9000 ð92; 192 15; 075ÞÞ 100 ¼ 8:7%: 10. Conclusion This paper begins with the technically useful part of heated moist air availability in the atmosphere, stratified in a gravitational field. Relying on other authors’ work (cited in our reference list), we accepted and elaborated on the possibility of a concentration of availability at ground level without a concrete chimney. Our proposed model’s main contribution relates to a thermodynamic approach that introduces a restriction of air return to the atmospheric stagnant pressure. Two possible solutions for solving the previously mentioned restriction are specified herein. One of the solutions is elaborated on, and it results in a simplified mathematical model of the process. This process is partially conducted in the solar collector and in ground level turbines. It is also vertically conducted through the troposphere as a flow in a gravitational vortex column (GVC). The rest of the process in the elaborated solution is connected with the heat rejection at the top of the troposphere and with power cycle closing. Typical integration results are shown by one characteristic numerical example of a GVC (as a part of a cycle) in the form of a table as well as in the form of a Mollier h–s diagram. Acknowledgements We would like to thank the anonymous reviewers and the Chief Editor for useful suggestions and recommendations. We would also like to thank the Croatian Ministry of Science, Education and Sports for funding this project. References Bernardes, M.A.S., Voß, A., Weinrebe, G., 2003. Thermal and technical analyses of solar chimneys. Solar Energy 75, 511–524.
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