On the Role of External and Internal Irreversibilities towards Classical Entropy Generation Predictions in Equilibrium Thermodynamics and their Relationship with Heat Transfer

In classical equilibrium thermodynamics description, ambiguities exist in terms of pinpointing external and internal irreversibilities in overall entropy generation prediction, as a system undergoes a thermodynamic process. The present work attempts to bridge this gap between classical thermodynamics-based irreversibility predictions and finite time heat transfer analysis. By choosing a model problem, expressions for external irreversibilities are quantitatively derived and it is observed that ambiguities may exist in the pertinent quantification depending on the very definition of system, immediate surroundings and surroundings. It is observed that if variations within immediate surroundings are taken into account, more realistic estimates of external irreversibilities can be obtained in equilibrium thermodynamics framework, as a function of heat transfer characteristics of the same. It is also shown that for some special cases, different expressions for external irreversibilities asymptotically converge to the same entropy generation predictions, in effect. INTRODUCTION As a consequence of the second law of thermodynamics, the sum total of change in entropy of the system and surroundings (i.e., for the Universe, in totality) is often expressed in equilibrium thermodynamics (classical) formulation as [1-3]: S2 S1 = Q T + Sgen 1 2 (1) where Q represents heat transfer across the system boundary (at local absolute temperature T) and S represents the entropy. In the above equation, subscripts 1 and 2 represent the end thermodynamic states for the process under consideration, and subscript ‘gen’ represents a generation (source) term that is supposed to be always greater than or equal to zero (principle of increase of entropy). Although in standard thermodynamics texts the above-mentioned entropy generation is generally mentioned as a consequence of irreversibilities [1-3], specific origin and direct quantitative measure of the same is nowhere pinpointed, leading to several ambiguities. For example, in many situations, it is not explicitly mentioned whether it is due to the total irreversibilities (internal + external) or internal irreversibilities alone. Even though scientific intuition clearly suggests that it is supposed to be a measure of internal irreversibilities within the system, the very notion of ‘system’ here may turn out to be quite confusing. Resolution of such anomalies is by no means an obvious task, since the fact whether external irreversibilities are absorbed within the term 2 1 T Q or not solely depends on *Address correspondence to this author at the Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302, India; E-mail: [email protected] the exact definition of the pertinent thermodynamic systems and their respective surroundings. In that context, ambiguities often arise in the notion of the terms ‘surroundings’ and ‘immediate surroundings’. Subtle differences between the various entropy generation predictions, therefore, may exist in accordance with the thermodynamic entities included in the ‘system’, ‘immediate surroundings’ and ‘surroundings’ themselves. Aim of the present work is to develop an unified approach in equilibrium thermodynamics framework to resolve the associated fallacies by outlining convenient means of pinpointing entropy generation due to internal and external irreversibilities, thereby, bridging some of the missing links between equilibrium thermodynamics and heat transfer. ANALYSIS In order to make a critical assessment of the parameters mentioned as above, let us refer to a model thermodynamic system (say, a substance enclosed in a piston-cylinder arrangement for example, refer to Fig. 1) interacting with its surroundings by means of total heat transfer Q across an area A. The heat is effectively transferred from the ambient (or equivalently, some other heat source) having a constant temperature of T . In order to reach the substance, thermal energy supplied by the source basically has to overcome two specific thermal resistances, namely, convective resistance between the ambient and the cylinder wall and conductive resistance within the wall of the cylinder itself (wall thickness = ). Consequently, the instantaneous temperature at ambient-wall interface (i.e., T0) and wall-substance interface (i.e., T) are different from the ‘surroundings’ temperature T . However, the substance comprising the thermodynamic system under concern can assumed to be at a state of thermodynamic equilibrium (i.e., change in state of the system is modelled by a so-called ‘quasiequilibrium’ or ‘quasi-steady’ process) with a consequence that at the thermodynamic state under consideration the entire system is in thermal equilib62 The Open Thermodynamics Journal, 2008, Volume 2 Suman Chakraborty rium as well. On the other hand, temperature differences are likely to exist across the two end faces of the cylinder-wall for the heat to be conducted across the same, and therefore, the wall is never in thermodynamic equilibrium. In fact, the very notion of thermodynamic equilibrium in the wall would preclude any possibility of heat transfer across it (since, heat is energy in transient by virtue of temperature difference). Therefore, if the wall is considered to be a part (or, an extension) of the system under investigation, it cannot be analyzed in the light of equilibrium thermodynamics, since thermal equilibrium necessarily demands a uniformity of temperature throughout the system at any thermodynamic state during the process. This might, however, give rise to ambiguous entropy generation predictions. In order to resolve the situation, we attempt here to make a combined ‘equilibrium thermodynamics’ and ‘heat transfer’ analysis of the situation, thereby pinpointing the nature of irreversibilities predicted through the statement of equation (1). In order to simplify the problem mathematically without losing pertinent physical features, we make the following assumptions for the present analysis: Fig. (1). A schematic diagram of the model situation considered for analysis. (i) The heat flux across the system boundary is timeinvariant. (ii) The behaviour of the system itself is transient, which, for simplicity, is considered to be a lumped mass of equivalent specific heat capacity C and mass m. Under the above assumptions, from energy conservation principles applied for the physical problem depicted in Fig. (1), we can write: 0 0 kA(T T ) dT Q Ah(T T ) mC dt = = = (2) where h is the convective heat transfer coefficient between the cylinder wall outer surface and the ambient. Although radiation heat transfer is not explicitly considered in the present analysis to avoid mathematical complications, it can be equivalently absorbed in the heat transfer rate equation by formulating an effective heat transfer coefficient heff such that 2 2 eff 0 0 h h (T T )(T T ) = + + + , being the Stefan-Boltzmann constant, and being the emissivity. Equation (2) can be conveniently rewritten as 0 0 T T T T T T dT mC / kA 1 / Ah / kA 1 / Ah dt = = = + (3) The above can be written in a compact form of: d mC 0 dt R + = (4) where T T = , R / kA 1 / Ah = + (overall resistance). Integrating equation (4) from an initial thermodynamic state (t=0, T=T1) to a final state (t=t, T=T) we get 1 t T T (T T )exp( ) mCR = (5)