ࡱ> mol_q` ^bjbjqPqP .::'eLLLLLLL`5558<55|`Np6h9~9~9~9:::MMMMMMM$OhQ@MLB::BBMLL~9~9NHLHLHLBL~9L~9MHLBMHLHLLLHL~9d6 z75EHLhM4N0NHL/RJ*/RHL/RLHL :`<HL>\?:::MMKv:::NBBBB```05```5```LLLLLL Manuscript title : THERMODYNAMICAL ASPECT OF DEFINITIONS ''CAPE'' AND ''TCAPE'' Author s: Ph.D. Neven Nini, associated professor Address: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, R.Boakovia b.b., 21000 Split, CROATIA mail : nninic@fesb.hr phone : 0038521305879 Fax:0038521305893 Dipl.ing. Zdeslav Juri, research assistant Address: Faculty of Maritime studies, University of Split, Zrinsko-Frankopanska 38., 21000 Split, CROATIA mail : mail : zjuric@pfst.hr mobil phone: 00385981653754 Dipl.ing. Sandro Ni~eti, research assistant Address: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, R.Boakovia b.b., 21000 Split, CROATIA mail : snizetic@fesb.hr phone : 0038521305881 mobil phone: 00385915696607 Fax:0038521305893 Sa~etak : U radu se analizira termodinami ki aspekt pojmova CAPE i TCAPE, definiranih na standardni na in i na na in u radu Rennoa; Ingersola (1996). Svi procesi koji su poslu~ili za definiranje tih pojmova ilustriraju se u termodinami kim dijagramima i opisuju to nim termodinami kim relacijama. U prvom slu aju oni su po definiciji potpuno ravnote~ni tj. reverzibilni. Nadalje, u radu se pokazuje da unutarnje trenje, koje Renno i Ingersol uzimaju u obzir, ini njihovu definiciju CAPE i TCAPE razli itom od standardne. Naime, po njihovoj definiciji CAPE i TCAPE predstavljaju rad disipacije pri konvekciji, a ne njen radni potencijal. Abstract: The paper analyses thermodynamic aspect of the terms CAPE and TCAPE, defined in both standard way and in the way done in Renno's, Ingersoll (1996) paper. All processes, which have served for defining these notions are illustrated in thermodynamic diagrams and described by accurate thermodynamic relations. In the first case, by definition this processes are reversible. Furthermore the paper shows that internal friction, taken in consideration by Renno and Ingersoll, make definition of CAPE different from the standard one. Namely, according to their definition, CAPE and TCAPE represent work of dissipation at convection and not its work capacity, which is not the same, either by physical content or numerically. 1. INTRODUCTION In the article Renno and Ingersoll (1996) convection processes are analysed as consequence of temperature non-equilibrium in the troposphere, situated between the ground of higher temperature and the stratosphere lower layer of lower temperature. The temperature non-equilibrium causes the occurrence of other forms of non-equilibrium, such as mechanical and concentration ones (e.g. change of water vapour content in the air). Out of the above-mentioned non-equilibrium, the mechanical non-equilibrium causes, initiates and maintains the convection motion in the atmosphere. Its numerical measure is at the same time the convection intensity value. In thermodynamics, this measure is presented by maximum work that can be obtained from the existing non-equilibrium. As, however, in circumstances of atmospheric convection it is possible to speak only of mechanical non-equilibrium bases, the maximum work reduces to adiabatic maximum work (Nini 2005). Such work that might be done by the low-level air in non-equilibrium atmosphere is in fact the convection intensity measure. In case when convection consists of raising the air from the ground, the maximum work equals the buoyancy force work. It can be obtained in an imaginary process of adiabatic equilibrium air rising to the maximum available height ''Hmax'' in particular circumstances. Therefore:  EMBED Equation.3  (1) By introducing specific volumes in place of densities, (1) gets the form well known in thermodynamics of flow processes:  EMBED Equation.3  (2) Namely, vdpa, is the convecting air flow differential work while rising up in the atmosphere, and -va dpa is part of that work "spent" on increasing its potential energy in the gravity field for static atmosphere (see further on following the equation (3)). The difference in (2) i.e.  EMBED Equation.3  is the work the air can perform besides rising, therefore creating the kinetic energy, i.e. the convection intensity. The designation  EMBED Equation.3  points out the adiabatic part of the updraft flow technical work capacity (i.e. its ''exergy'').  EMBED Equation.3 further on referred to as updraft work capacity. 2. CONVECTION INTENSITY MEASURE IN METEOROLOGY In the definition of low-level air technical work capacity, first proposed in meteorology by Brunt (1941), the updraft movement conditioned by downdraft, also having its technical work capacity, is taken in consideration. We need to point out that downdraft movement of air is created and maintained by rejecting heat to the height Hmax. Because of rejecting the heat, the air which from the ground comes to the height Hmax, gains a new "buoyancy force", however in the opposite direction (downward), due to its change of state. If the updraft work capacity  EMBED Equation.3  according to (2) (in meteorology called CAPE) is added the downdraft i.e.  EMBED Equation.3 , (or DCAPE calculated analogously with (2)), updraft-downdraft or total work capacity TCAPE can be obtained. If there was no separation of condensed phase, the sum of the two work capacities  EMBED Equation.3  and  EMBED Equation.3 could be replaced by work of Carnot reversible cycle between temperatures at the ground and at the maximum height. It is the maximum work based on temperature non-equilibrium as the primary one. The total work capacity was first derived by Brunt in the example of air movement as an ideal gas in the Carnot cycle. The updraft-downdraft moving air in fact performs the Brayton cycle, equivalent to the Carnot. It consists of two adiabatic curves (updraft or "warm" and downdraft or "cold") and two isobars. The first isobar is so called "near-ground" one, during which the air receives heat from the ground. The second isobar is at the height Hmax, during which the air transmits the heat lowering its own temperature in the surrounding atmosphere at H=Hmax. All these processes are shown in solid lines 1234 in p-v diagram in Figure 1. Fig. 1 The near-ground air state is presented by point "1". Process 1-2 represents the equilibrium adiabatic expansion where the air rises up to Hmax. In this case T2 > T3 = Ta(Hmax). The convecting air delivers heat at that height, in which process it cools to T3 = Ta(Hmax). By equilibrium adiabatic downdraft (process 3-4), the convecting air returns to the near-ground pressure, to state 4. Points 1 and 3 show the surrounding atmosphere states in the diagram in Figure (1), i.e. the dotted curve 1-3 represents the relationship per height for the atmosphere states. The latter is in accordance with the fact that the atmosphere must obviously be unstable. Let us now interpret in Figure (1) the notions regarding the definitions  EMBED Equation.3  and  EMBED Equation.3 , i.e. CAPE and DCAPE according to Brunt (1941). The area AA13B, represents potential energy at the height Hmax, i.e.  EMBED Equation.3  (3) The evidence for this is the following. Let us take 13 as imaginary equilibrium diabatic flow process, with air rising. In such case the area AA13B, will represent the technical work done by that air. As the process was without buoyancy force, with v (z) = va (z), the whole work was spent on increasing its potential energy. Thus, if the total warm air technical work at its rising represents the area AA12B, then the triangle shape area A123, represents the work which is the updraft air flow intensity measure, which is also contained in (2), i.e.:  EMBED Equation.3  (4) In the Appendix, there is a new and simple CAPE numerical calculation method for humid air. The method works with the real moist air and arbitrary density distribution per height a (z). Analogously (4), the surface A413 in Figure 1 represents the downdraft movement work. This work is at the same time the downdraft movement intensity measure. Therefore, if the convection intensity at a location is influenced by updraft and downdraft flow, it is then justifiable to take TCAPE or instead of (2) i.e. CAPE:  EMBED Equation.3  (5) Although Brunt (1941) introduces TCAPE based on the Carnot instead of more realistic Brayton cycle in Figure 1., actually there is no essential difference if they are, as already said, equivalent. Namely, if the Carnot cycle lower and upper temperatures are mean thermodynamic temperatures corresponding to the Brayton cycle isobares, then between them there is no difference in efficiency. 3. ENERGY DISSIPATION INFLUENCE Emanuel (1986, 1989) touches on the interesting problem of mechanical energy dissipation influence to dissipation intensity itself. Namely, the CAPE definition according to (2) can be maintained, provided that pressure function v under the integral is substituted by v-function in a real process, which takes in consideration energy dissipation. Let us dwell on now on such alternative understanding of CAPE* and TCAPE* which we mark with asterisks and which, to a certain extent, differ from the original according to Brunt. Unlike Brayton and Carnot cycle, the one with internal friction in Renno and Ingersoll's paper (1996) is internally non-equilibrium. As a modification of equilibrium Brayton cycle, that cycle is achieved so that Brayton cycle is added internal friction i.e. mechanical energy dissipation. The internal friction is contained in adiabatic compression and expansion processes. By application of the above mentioned modifications the equilibrium Brayton cycle transits into cycle 12*34*, Fig.(1). and Fig.(2). Approaching thermodynamic analysis of such a (real) cycle of the convecting air, we start from the First Law of thermodynamics, and this for an observer connected to the Earth. As for this observer there is no transfer of work with external bodies, for all process parts, the first law boils down to the following:  EMBED Equation.3  (6) with  EMBED Equation.3  (7) Here, q is added heat, and h is humid air specific enthalpy. The value htot represents the total humid air specific energy of fluid flow. Enthalpy h is specified by its temperature, pressure and moisture content per 1 kg of pure air r. The law on conservation of energy (6) is applicable on both equilibrium and non-equilibrium processes. Non-equilibrium processes with internal friction, a subject matter herein, represent a particular case. Such non-equilibrium processes can be modelled with the equivalent equilibrium processes, according to the method hereinafter presented. According to this method, the friction work appears explicitly in (6) in the form of two additional terms. The first such term is added to the right side of the expression (6) and represents fictive reversible work. This work is numerically equal to the internal friction work dwfr. The other supplementary term is added to the left side (6) and represents added heat at the same rate dqfr. Therefore, in the equilibrium model of a non-equilibrium process equation (6), is transferred to:  EMBED Equation.3  (8) where dwtr is the internal friction work expressed in equilibrium form  EMBED Equation.3  (9) with  EMBED Equation.3  (10) where dpfr and dsfr are pressure and entropy differential changes due to the friction process. According to the same equilibrium model, this time for the thermodynamic observer connected to a small portion of moving air:  EMBED Equation.3  (11) Treatment of real processes is expressed by the equations (8) to (11) is known in technical thermodynamics. The first author himself has some additional arguments concerned the application of this modelling method. Renno and Ingersoll (1996) according to Emanuel (1986, 1989) substitute in (7) the approximate expression for humid air enthalpy:  EMBED Equation.3  (12) in accordance with (8) and (10) writing:  EMBED Equation.3  (13) with  EMBED Equation.3  (14) 4. AlternativE EXPRESSION FOR TCAPE Integrating (14) and with (11) along real convection updraft - downdraft cycle and taking in consideration that h is a state function - the following is derived:  EMBED Equation.3  (15) Integral -  EMBED Equation.3 represents work of potential energy and kinetic energy changes and internal friction per cycle. As for a cycle a potential and kinetic energy zero, then:  EMBED Equation.3  (16) This is in accordance with (13) and (15). Formally, it is also easy to present  EMBED Equation.3  (17) as in any case cycle areas in rectangular p-v and T-s diagram, are always equal. The latter (17) coincides with Renno and Ingersoll's (1996) equation (3). The cycle it is valid for 12*34*, is not the same as the cycle 1234 in Figure 1, where CAPE and TCAPE are defined and interpreted graphically. In Figure 2 there is the alternative presentation of both cycles in T-s diagram. Fig. 2 According to the model supporting the equations, (8) and (14) all real processes may be formally considered internally equilibrium. According to (16) and (17) all friction work done returns to the air as "heat" qfr in the processes 12* and 34*, and the total ''added'' heat equals:  EMBED Equation.3  (18) Rejected heat is at the same time actually rejected  EMBED Equation.3  (19) while actually added heat is only:  EMBED Equation.3  (20) Friction work is therefore  EMBED Equation.3  (21) with  EMBED Equation.3  i.e.  EMBED Equation.3  (22) According to the same model and the equations (16) and (17), friction work is the real cycle area in the rectangular T-s and p-V diagram. The value wfr.cy presents real friction work during convection cycle, and obviously differs from supposed TCAPE according to (5). Physical interpretation of this difference is as follows: in the processes with internal friction there occurs loss of work or mechanical energy dissipation, and, at the same time there is the increase of the air technical work capacity in the continuation of the process (due to the existence of "added heat" from internal friction work-reheat factor). For the previously stated reasons, the expression (17) occurring with Renno and Ingersoll (1996), represents the convective cycle actual work dissipation,  EMBED Equation.3  (23) However, TCAPE does not represent actual friction work, but total work capacity for energy dissipation. It can only be pre-calculated out of the unstable atmospheric conditions. Renno and Ingersoll (1996) introduce TCAPE* under the name TCAPE and give it the sense for a boundary layer convection as a heat engine, having in mind the cycles with humid air without precipitation (below cloud base). They extend the same conclusions to a more general case of deep convection as a heat engine. The same authors, however, do not make the difference herein emphasised between TCAPE* and the standard meteorological definition TCAPE. They, also, interpret TCAPE* according to (23) as a magnitude derived for a reversible heat engine, which obviously is not the case, as the internal friction in the equations (13) to (23) is incompatible with reversibility. 5. CONCLUSION Analysing the thermodynamic aspect of convection intensity, we can conclude that the integral in (23) according to Renno and Ingresoll (1996) represents the actual friction work in a convection cycle, i.e. mechanical energy dissipation. The authors do not point out the difference according to the standard meteorological definition TCAPE, or its physical basis. They call the real cycle with internal friction equilibrium one, which actually it is not. According to the presented in this analysis, TCAPE* is not a criterion for the evaluation of convection intensity based on the atmospheric state preceding or accompanied by during the convection. It is only the dissipation actual work magnitude and incalculable as the actual process - symbolised by the states 2* and 4* in the diagrams - is not known in advance. The convection intensity criterion for given atmospheric conditions must be total work capacity, therefore reversible, i.e. without internal friction. In a sense, the standard meteorological definition CAPE and TCAPE is better, and besides, it is unambiguously determined by the atmospheric conditions. The new, alternative procedure of CAPE calculus presented in the Appendix has not been compared with the existing calculation methods. It is particular for fully taking in account the actual characteristics of humid air, it is physically transparent, and besides the h-r diagram, mere use of calculator is sufficient. NOMENCLATURE:  EMBED Equation.3  - maximum work of downdraft air  EMBED Equation.3  - maximum work of updraft air Hmax [m] - maximum height (at troposphere level) a [kg/m3] - density of atmospheric air  [kg/m3] - density of convecting air v [m3/kg] - specific volume of air q [J/kg] - specific heat z [m] - height p [Pa] - pressure A - area in thermodynamic diagram  EMBED Equation.3  - specific potential energy CAPE  EMBED Equation.3  - convective available potential energy TCAPE  EMBED Equation.3  - total available potential energy DCAPE  EMBED Equation.3  - downdraft available potential energy htot [J/kg] - specific total enthalpy h [J/kg] - specific enthalpy v [m/s] - intensity of speed w [J/kg] - specific work s [J/kgK] - specific entropy cp [J/kgK] - specific heat capacity of the humid air g [m/s2] - acceleration of gravity T [C] - temperature r [kgv/kgd.a.] - mixing ratio (in general including water vapour, liquid and ice ) L[kJ/kg] - latent heat R [J/kmolK] - universal gas constant Subscript: a - atmospheric fr - friction cy - cycle in - input out - output cond - condensation v - vapour n - new numerical step p - previous numerical step APPENDIX CAPE Calculus Based on the Humid Air State Diagram Direct numeric CAPE calculus according to (2) is possible without any simplifications of humid air characteristics, if five individually simple calculus steps are combined. The first three steps are directed to finding the connection of the convecting air specific volume with the pressure at its adiabatic rising. It is the function v = v (pa) in (2). The fourth step regards subtraction of known atmospheric height profile va (pa). This for steps show changes of all state properties over one rising stage. The final step is summarising the differences v (pa) va (pa) overall rising stages as integral approximation in (2). The first three steps are determined by means of the Mollier h - r diagram for moist air, Bonjakovic F., Blackshare P. L. (1965), or what is fully equivalent to this diagram psychrometric chart Group of authors (1997). Although any state of air is determined by three magnitudes (t, r and p = pa (z)), it is a unique diagram in which a state is determined by a point (t, x), plus one parametric curve dependable on pressure. This dependency is so simple that having h-r at your disposal just for one (standard atmospheric) pressure, to be able to adjust the same diagram to any other pressure. This is achieved by a short conversion, which only moves the parametric curve - line of humid air saturation. Besides the Mollier h-r diagram for finding CAPE, it is necessary to know the function v = v (t, r, pa):  EMBED Equation.3  (i) where: rv content of humidity as vapour, rcond content of humidity as condensed phase, and R universal gas constant. The first step in the calculus v (pa) at equilibrium adiabatic air rising is the relation  EMBED Equation.3  , (ii) derived from (11) for equilibrium adiabate. The calculus begins from the given near-ground air state, where the pressure changes with the adopted numerical step pa. 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After the first such step the changed enthalpy is known hn = hp + h and the new pressure pan = pap + p, while the humidity content r remains the same. The above said is the same in case of vapour condensation occurrence, if the condensate does not precipitate from the air. The second step is finding the new state, as a point in h-r diagram for the given h, r and pa. Reading from the diagram all the necessary in (i) data are gathered for the third step calculus - namely, for the new specific volume calculation. There follows the repetition of the same three steps for the following rising stage as the new state etc. for all lower and lower pressures, i.e. for all higher co-ordinates z. After reaching the pressure at the maximum height pa min = pa (Hmax) the obtained dependency v (pa), together with the given function va (pa), is substituted in the sum replacing the integral in (2) and further on CAPE. The procedure for air lowering and DCAPE calculus is analogous. The whole procedure is adjusted for computer application, for which it was only necessary to build in the h-r diagram into the memory and make the whole described procedure algorhitmic. REFERENCES: Brunt, D., 1941 : Physical and Dynamical Meteorology. Cambridge University Press, pp. 428 Bonjakovic F., P. Blackshear, 1965: Technical Thermodynamics. Holt, Rivehart and Winston, 524 Emanuel, K. A., 1986 : An air-sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. Journal of Atmospheric Sciences, Vol. 43, 585-604 Emanuel, K. A., 1989: Polar lows as arctic hurricanes. Tellus, 41A, 1-17 Ninic, N., 2005 : Available energy of the air in solar chimneys and the possibility of its ground-level concentration. Solar Energy (in the press) Renno O. N., Ingersoll A. P., 1996 : Natural Convection as a Heat Engine. Journal of Atmospheric Sciences, Vol. 53. No. 4., 572-585. Group of Authors, 1997: ASHRAE Handbook-Fundamentals, ASHRAE Inc. FIGURES :  Figure 1. Brayton equilibrium cycle of updraft-downdraft air movement  Figure 2. 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