Proceedings of ITP2007, Paper Number ITP-07-20, Interdisciplinary Transport Phenomena V: Fluid, Thermal, Biological, Materials and Space Sciences October 14-19, 2007, Bansko, Bulgaria COMPUTATIONAL AND GROUND BASED EXPERIMENTAL INVESTIGATIONS OF THE EFFECTS OF SPECIFIED AND UNSPECIFIED (FREE) PRESSURE CONDITIONS AT CONDENSER EXIT FOR CONDENSING FLOWS IN TERRESTRIAL AND MICRO-GRAVITY ENVIRONMENTS 1 Amitabh Narain , Shantanu Kulkarni, Soumya Mitra, Jorge H. Kurita and Michael T. Kivisalu Department of Mechanical Engineering-Engineering Mechanics Michigan Technological University Houghton, MI-49931 ABSTRACT Reported experimental and computational results confirm that both the flow features and heat transfer rates inside a condenser depend on the specification of inlet, wall, and exit conditions. When exit pressure is changed from one steady value to another, the changes required of the interior vapor flow towards achieving a new steady duct flow are such that they do not demand removal of the new exit pressure imposition back to the original steady value. This behaviour of condensing flow is unlike that of any incompressible single phase duct flow which cannot allow imposition of an exit pressure other than the “required” value. Instead, new steady flows may be achieved for different steady exit pressure prescriptions (termed category I flows) through appropriate changes in the vapor/liquid interfacial configurations and associated changes in interfacial mass, heat transfer rates (both local and overall), and other flow variables. This special feature of these flows is for the commonly occurring large heat sink situations for which the condensing surface temperature (not heat flux) remains approximately the same for any given set of inlet conditions while exit condition changes. In this paper’s experimental context of flows of a pure vapor that experiences film condensation on the inside walls of a vertical tube, the reported results provide important quantitative and qualitative understanding. The theorotical and experimental results presented in this paper allow us to propose important exit-condition based categorization of these flows. Of these, category II flows, which are defined to be the cases for which exit pressures are left unspecified, have typically been assumed to hold for most actual steady realizations of gravity driven flows. However it is shown here that steady flows under self selected exit pressure conditions are achieved under these conditions only if special hardware arrangements are made that allow vapor to freely choose an exit pressure. If this is not so, one often has an inadvertent category I flow without the explicit knowledge of the imposed exit pressure condition. It is theoretically shown here that though this category II flow situation in the absence of specification of exit condition constitutes an “ill posed” steady boundary value problem, steady flows may still be experimentally realized under special arrangements. This fact is also validated by unsteady theory which shows that solutions from many different initial conditions may tend to a unique steady attracting solution (as t ) for some of the internal and/or external flows. These “attractors” exist and are strong for gravity driven flows and are often weak to non-existent for zero gravity or horizontal shear/pressure driven flows (both for internal and external flows). In microgravity, the “attractors” for category II internal flows are weak in the sense that they significantly lose their strength of “attraction” after a certain downstream distance – after which the flows are indeterminate if exit pressure remains unspecified. For these microgravity situations, the remedy is to run condensers under suitably specified inlet and exit pressures (category I conditions) as well as a proper cooling strategy (i.e. proper wall temperature variations). Key-words: film condensation, phase-change heat transfer, two-phase flows, interfacial waves. 1 To whom correspondence should be addressed. E-mail: narain@mtu.edu, Phone: 906-487-2555 Fax: 906-487-2822 1 NOMENCLATURE Cp1 Cp2 D G hfg(pin) h h Ja Ja|V k1 L LC M in ML MV pB pD2 pT" pin pexit Pr1 pxi q′′ Qb Q out Re TB TR Tsat(p) Ts-xi Tw Tw( ) TV-in TC-in u U v x xe Specific heat of the liquid condensate, J/(kg-K) Specific heat of the vapor, J/(kg-K) Inner diameter of the test-section, m Inlet mass flux, 4· M / (π·D2), kg/(m2/s) Heat of vaporization at pressure pin, J/kg Average heat transfer coefficient, Qout / (π·D·L), W/(m2-K) Channel gap, m Condensate Jakob number, Cp1· ∆ T / hfg(pin) Vapor Jakob number, Cp2 Tsup/hfg(pin) Conductivity of condensate liquid, W/(m-K) Length of the test-section, m Characteristic length, LC = D for tubes and LC = h for channels, (m) Vapor flow rate at test-section inlet, g/s Liquid flow rate at test-section exit, g/s Vapor flow rate at test-section exit, g/s Evaporator (boiler) pressure, kPa Pressure at the location near the rotameter, kPa Pressure at a location downstream of the test section, kPa Pressure at the test-section inlet, kPa Pressure at the test-section exit, kPa Condensate liquid Prandtl number, µ1·Cp1 / k1 Test-section pressures at different locations = i (i = 1, 2, …), kPa Average convective heat flux, W/m2 Net heat rate into the evaporator, W Net heat rate out of the test-section, W Inlet vapor Reynolds number, 4· M /(π·D·µ2) Evaporator fluid temperature, oC Rotameter fluid temperature, oC Saturation temperature at pressure p, oC Condensing surface temperatures at different locations = i (i = 1, 2, …), oC Mean condensing surface temperature, oC Non-uniform steady condensing surface temperature, oC Vapor temperature at test-section inlet, oC Temperature of the counter-current coolant water flow at the approach to the testsection, oC Non-dimensional velocity in the x-direction Velocity in the x-direction, m/s Dimensional velocity in x-direction, m/s Non-dimensional velocity in y-direction Dimensional velocity in y-direction, m/s Non-dimensional distance in x direction ( /LC) Ratio of test section length to characteristic length (L/LC). in in 2 fc y ∆T Tsup p Ze Z( ) δ 2 1 2 1 D Distance in x direction, m Approximate length needed for full condensation (estimted by computations), m Non-dimensional distance in y direction ( /LC) Distance in y direction, m Tsat(p) - T w , oC Vapor superheat, TV-in - Tsat(p), oC pin – pexit, kPa Ratio of exit vapor mass flow rate to total inlet mass flow rate Ratio of vapor mass flow rate to total mass flow rate at any location x along the test section Non-dimensional value of condensate thickness Density of vapor, kg/m3 Density of liquid, kg/m3 Viscosity of vapor, kg/(m-s) Viscosity of liquid, kg/(m-s) Transient decay time for disturbances, s Subscripts Exit In Na Exp comp Test-section exit Test-section inlet Natural steady case Obtained from experiments Obtained from computations 1. INTRODUCTION AND BACKGROUND This paper presents a synthesis of results obtained from a fundamental and novel experimental investigation (see Narain et al. [1]) of effects of exit conditions on internal condensing flows and computational investigations ([2]-[6]) of internal and external condensing flows. In section-6, the paper also presents a new set of generalized results for this and other internal two-phase flows. Reported experimental results confirm the existing computational results for internal condensing flows ([4], [5], [6]) that both the flow features and heat transfer rates inside a condenser depend on exit pressure conditions over and above inlet and wall conditions (i.e., the method of cooling for the condensing surface). The steady condensing surface temperature (not heat flux) variation is assumed fixed and known or knowable (through consideration of the appropriate conjugate problem). This paper identifies and establishes a multiplicity of steady/quasi-steady solutions – and/or oscillatory flows – under different steady conditions at the exit. Experiments support the existing simulation results ([4], [5], [6]) that have already shown, among other results, the presence of multiple steady/quasi-steady solutions under multiple steady specifications of the exit condition. The condenser exit condition (for partial as well as full condensation) is specified by exit pressure. For partial (incomplete) condensation flows, specification of exit pressure is equivalent (see Fig. 6 in [4]) to specification of exit vapor quality i.e. the ratio of vapor mass flow rate at the exit to the inlet mass flow rate (see simulation results from [4]-[6]). The computational simulations ([4], [5], [6]) for the gravity driven partial condensation cases (such as flows inside a vertical tube) also predict that, for a certain set of inlet and wall conditions, even if the exit condition is not specified and a suitable range of exit conditions is available for the flow to choose 3 from, the condensate flow under gravity makes the overall flow seek and attain a specific “natural” flow and associated exit pressure condition (in Narain et al. [4] this situation is also termed as a natural unspecified steady exit condition due to the presence of an “attractor,” i.e., an “attracting” solution). Unfortunately many planned system designs incorporate a condenser and assume that the condenser will always find a “natural” steady flow since no exit conditions are specified. This is not generally true. The attainment of natural steady flows under unspecified exit conditions require special hardware arrangements and, even then, they occur more readily (over a larger parameter zone) for gravity driven condensate flows than they do for shear driven flows (see [2] – [3] and [6]). The reported experimental results for gravity driven partial condensation cases under unspecified exit conditions support the computational results – both qualitatively and quantitatively. Therefore, for an unspecified exit condition flow, the steady “natural” exit condition may or may not exist depending on whether or not a steady “attractor” exists. For example, as shown in [6], an attractor does not exist for many slow to moderate inlet flow rates in horizontal or zero gravity situations. Furthermore even if an “attractor” does exist, its realization depends on whether the attracting “natural” exit pressure value falls within the range of exit pressures made available to the condensing flow test-section by the rest of the system/flow-loop. In the absence of active specification of a steady exit condition, the available range of steady exit conditions is determined by the components downstream of the condenser as well as the specific nature of the hardware used in the design of the rest of the system (or flow loop). Furthermore, because of the small pressure drops (see [6] or experimental runs reported here), when a steady “natural” exit condition is achieved under unspecified conditions, or when the specified exit condition is not too far from this “natural” steady exit value, typically, vapor flows are typically close to incompressible. The vertical in-tube internal partial condensing flows are investigated here for a downflow configuration. Though numerous in-tube condensation experiments have been done, most of the well known in-tube vertical downflow experiments done by Goodykoontz and Dorsch [7]-[8], Carpenter [9], etc. either limit themselves to sufficiently fast flow portions of the duct that do not significantly depend on exit conditions or operate under a particular (natural or otherwise) set of exit conditions (that gets specified or remains unspecified depending on the employed experimental set up) and, therefore, results may vary from one experimental system to another. In addition to our group’s very early (see Yu [10]) and subsequent ([4], [5]) computational findings on the importance of exit conditions for internal condensing flows, experimental findings of Rabas and Arman ([11]) have also indicated the significance of exit conditions through their observation that the presence or absence of valves at the exit affected some of their in-tube vertical downflow results. Furthermore, several experimental results and analyses ([12] - [19]) indicate that, for certain physical arrangements leading to a specific class of inlet and outlet conditions, transients and instabilities are expected in complete condensation horizontal (or slightly tilted) in-tube internal condensing flows. Since these experiments and the corresponding modeling techniques in the literature limit themselves to a particular type of inlet and exit conditions, they do not directly apply to the presence or absence of observed transients and instabilities in other feasible categories of exit-condition specifications. In the context of boundary value problems for internal condensing flows, what is new is that our computational and experimental results unequivocally show that the commonly occuring condensing flows are very sensitive to the nature of exit conditions as well as to the changes in exit conditions (due to changes in exit pressure). This sensitivity arises from the ease with which these changes alter the vapor flow field in the interior. Therefore when only exit pressure condition is changed from one steady value to another, the changes required for the interior vapor flow towards attaining a new steady flow 4 are such that they do not demand removal of the new exit pressure imposition - as is the case for incompressible single phase duct flows with only one allowed exit pressure. Instead, for condensing flows, new steady flows are achieved for new exit conditions through appropriate changes in the vapor/liquid interfacial configurations and associated changes in interfacial mass, heat transfer rates (both local and overall), and other flow variables. This happens because the “interface tracking equation” (see, e.g., eqn. (20) of [4] which arises from setting the interfacial mass flux L K given in eqn. (A.10) of the Appendix equal to another interfacial mass-flux Energy given in eqn. (A.11) of the Appendix) which locates the interface involves a key parameter Energy . This key parameter Energy which is obtained from the interfacial heat flux associated with the local temperature gradient easily changes with changing interface location; particularly if the interface is near a heat transfer surface at a prescribed wall temperature. It is the absence of an available adjustable range for interfacial mass-flux Energy for single-phase flows (which have no interface) and for adiabatic gas-liquid flows (where this interfacial mass-flux is constrained to be zero) that makes it difficult to externally impose different exit pressures for these better understood flows. Fig. 1: The schematic for the flow through test-section and exit condition issues. Fundamentally, internal condensing flows’ exit condition sensitivity arises from governing equations being “elliptic” or requiring “two-way” space co-ordinates in the flow direction (i.e. flow variable at a point P is influenced both by upstream and downstream local neighbors as shown in Fig. 1c) to determine the flow fields in the two phases and the interface location. However, this sensitivity to exit 5 conditions, or “two-way” behavior, is special and does not result from the typical “ellipticity” associated with slow flows and flow reversals (which are, as described in Patankar [20], associated with the changes in the sign of local Peclet numbers). In fact, it is found (for simulations in [4]-[6]) that even when vapor and liquid flows are unidirectional (or “one-way” or “parabolic”) due to local Peclet numbers (which appear in local discretization equations for the velocity components - see eqs. (5.61) – (5.64) in [20]) being very large (greater than 50,000 in both x and y directions for discretizations used in exit-condition sensitive simulation results reported here), the flows exhibit “elliptic” or “two-way” behavior leading to sensitivity to exit conditions. This special sensitivity to exit condition is due to “twoway” behavior of the vapor pressure fields (reflected, in the context of our computational methodology [4], by the “two-way” behavior of the pressure equations given by eqs. (6.30) – (6.31) in [20]) combined with the degree of freedom available with regard to relocation of the interface towards accommodating changed vapor flow fields through changed interfacial mass-fluxes. These equations for condensing flows are such that the coefficients that multiply the pressures at the locally upstream and downstream neighbors are comparable even for large Peclet numbers (see pressure discretization equations for SIMPLER procedure in Patankar [20]). Due to this, effects of changes in the exit pressure are felt by the entire vapor flow field which is then able to accommodate this change by changing the interface location and achieving a new flow field with different interfacial mass-transfer rates (and hence heat-transfer rates). A much simpler support for the above result comes from simple one-dimensional modeling (leading to ordinary differential equations) of laminar/laminar internal condensing flows in a channel. It is appropriate to ignore the integral form of vapor momentum balance only for the gravity dominated external flow problem of Nusselt [21] and, occasionally, for some gravity driven channel flow analyses. For gravity driven internal (e.g. channel) flow analyses this approximation means condensate motion is not affected by vapor momentum balance because of negligible interfacial shear and freedom of the vapor to choose its exit pressure – which is a freedom that does not always exist. If one does not ignore the integral form of vapor momentum balance, the correct formulation of the problem is achieved (see, e.g., eqn. (53) of [22] for a horizontal channel). The ordinary differential equations for the correct formulation involves first order derivatives for other flow variables (interface speed uf(x) and film thickness δ(x) in [22]) but second order derivatives (d2π/dx2) of the variable π(x) that represents the nondimensional value of cross-sectional pressure. To solve this set of equations correctly, which was not done in [22], one needs exit pressure specification over and above specifications of inlet (pressure, velocity profile, and film thickness values) and wall temperature conditions. At the time of the analysis reported in [22], though eqn. (53) in the formulation given in [22] was correct, the solution was not. Instead of solving the equations for different exit pressures, the reported solution effectively chose only one exit pressure by an ad hoc mathematically specified value for the pressure gradient (dπ/dx) at the inlet. In hindsight this was done to conform to the popular notion that the flow should be parabolic, even though the analytical formulation in [22] suggested that the flow is “elliptic” and requires exit pressure prescription. A serious problem with this type (as in [22]) of correct one dimensional steady formulation is that it always requires exit pressure specification as a boundary condition. Such an analysis is incapable of identifying the cases where exit pressure prescription is not required, such as fast partially condensing flows in ducts of small lengths or gravity driven steady flows in a vertical channel under unspecified exit conditions. In these cases, if the flow is allowed to choose its own exit pressure, it should self select its own “natural” exit pressure. The results based on unsteady solutions for these situations, which are presented here, exhibit presence of an “attracting” solution for the mathematically “ill posed” steady formulation. These results assure us that there exists a steady solution even in the 6 absence of exit pressure specification and the associated “ill-posed” steady formulation. There are other types of one-dimensional parabolic flow formulations/analyses that are frequently employed to allow turbulent vapor or wavy interface through introduction of an independently assumed model for “interfacial friction” Si. This interfacial shear Si relates to laminar liquid velocity profile through first equality in eqn. (A.7) of the Appendix but does not require satisfaction of the second equality in eqn. (A.7) of the Appendix – this is because vapor flow is allowed to be turbulent near the interface. The integral momentum balance for the vapor directly uses the interfacial friction Si. Such steady formulations involve assumed “models” for interfacial shear Si (see, e.g., Narain et al. [23]) and they work because the model functions are such that they can yield known values of Si during forward marching of the solution through a first order system of non linear ordinary differential equations under prescribed initial conditions at the inlet. As shown later, with the help of the complete solution of the full two-dimensional steady/unsteady problems, results obtained from one dimensional theory employing such models can be good only over finite length problems involving partial condensation and sufficient vapor velocities (with either laminar or turbulent vapor flows). Therefore, these models are fundamentally flawed and need to be changed to allow second order derivatives in pressure (as is the case in [22]) if one is looking at complete condensation problems or sufficient downstream distances where vapor flows are slow and are sensitive to exit pressure. Fig. 2: For tube flow situations specified as in Phan et al. [6] (see their Table 1 and Fig. 2), the figure depicts three steady film thickness profiles (curves C1 to C3) for three different exit conditions for category I steady flows. The same figure is also used to show results for category II gravity driven flows once the exit condition specification is removed and the flow is allowed to seek its own exit condition (curve C4). For category II flows, the figure indicates time trends for two sets of δ(x,ττ) predictions for τ > 0. The curve C1 starts at Ze = 0.3 at τ = 0, and, as τ → ∞, tends to the curve C4 which represents the solution for Ze = Ze|Na = 0.215. The curve C2 starts at Ze = 0.15 at τ = 0 and tends, as τ → ∞, to the same curve C4. The curve C3 represents category I solution when exit condition is held fixed exactly at the natural exit condition. As expected, the curves C3 and C4 overlap each other. In the literature, condensing flows have been classified as to whether they are shear dominated or gravity dominated, internal or external, smooth or wavy at the interface, laminar or turbulent in the two phases, etc. It is proposed here that one can only make sense of the vast literature on forced internal 7 condensing duct flows if they are also classified in different categories based on the conditions imposed at the inlet and the exit. The following three categories (termed categories I – III) proposed here cover most cases of interest. Category I (with complete or incomplete condensation under specified exit conditions) • Prescribed or known values of total inlet vapor mass flow rate Min (kg/s), inlet vapor quality, inlet pressure pin, and inlet temperature at all times t. Without loss of generality, one can focus on an all vapor flow (an inlet vapor quality of unity) at the inlet with known values of total inlet (all vapor) mass flow rate M in (kg/s), inlet pressure pin, and inlet temperature (at saturation temperature Tsat(pin) or at some superheat) at all times, t. This means the prescription could be steady or unsteady. • Prescribed method of cooling that leads to known condensing surface (wall) temperatures Tw( ,t) < Tsat(pin) for all -locations over which film condensation occurs. Typical wall temperature conditions of interest are steady, but unsteady conditions are relevant to start-up and shutdown. • Specified or known exit pressure conditions. For example, for steady exit conditions, exit pressure pe = constant, which is equivalent, for steady partial condensation flow cases, to setting exit mass quality Ze ≡ M V (kg/s) M in (kg/s) equal to an appropriate constant, where M V (kg/s) is the vapor mass flow rate at the exit. For category I partial condensation flows with steady specified exit pressure and other conditions, the computational simulation results shown in Fig. 2 are the solutions of the formulation given here in the Appendix. The flow conditions and formulation are the same as the ones specified for Fig. 2 and Table 1 of Phan et al. [6]. The results show three different steady solutions (curves C1 to C3) for three different specified vapor qualities Ze at the exit (viz. Ze1 = 0.3, Ze2 = 0.15 and Ze3 = 0.215). Some of the Garimella et al. experimental investigations ([24]-[25]) are for flows in this category with inlet quality Zi (< 1) and exit quality Ze specified at values incrementally smaller than Zi. Category II (with complete or incomplete condensation under unspecified exit conditions) • Prescribed or known values of inlet mass flow rate M in (kg/s), inlet pressure pin, and inlet temperature (at saturation temperature Tsat(pin) or at some superheat) at all times, t. Without loss of generality, it is assumed that the flow is all vapor at the inlet. • Prescribed method of cooling leading to known wall temperatures Tw( ,t) < Tsat(pin) for all locations over which film condensation occurs. • Exit condition is not specified. However, due to some system hardware limitations and constraints, if only a range of exit conditions (i.e. pressures) is available and the condenser is freely able to choose its “natural” exit pressure within this range, we still call it a category II flow. On the other hand, in the case of the availability of only a limited range of exit pressures, if the flow seeks a “natural” value that is at or outside the boundary of the limited range, the flow is not considered a category II flow. 8 - For category II unsteady partial condensation flows with unspecified exit pressure or exit quality, the computational simulation results shown in Fig. 2 are the solutions of the formulation given here in the Appendix. The steady formulation for this problem is however “ill-posed.” The flow conditions and formulation are the same as the ones specified for Fig. 2 and Table 1 of Phan et al. [6]. For these gravity driven condensate cases of category II flows shown in Fig. 2, it is computationally shown that if exit vapor quality specification constraints at Ze1 = 0.3 or Ze2 = 0.15 are removed at some time (τ = 0) – and subsequently (τ > 0) one does not specify any exit condition in the unsteady mathematical model (which is equivalent to not specifying exit pressures and allowing them to vary over a certain range), the solutions show that the experimental flow in this category should be able to select, in accord with the predictions, a quasi-steady flow with a “natural” value of exit quality (curve C4, representing Ze3 = Ze-Na = 0.215). However, as discussed later, experimental arrangements for category II flows require special design and the available range of experimental conditions may not include the “natural” exit condition the flow is seeking. It is also possible that natural “attractors” for category II flows are weak or do not exist, as is shown to be the case for some horizontal and zero gravity condensing flows discussed later. For these cases, the concave bowl analogy schematic for “attractors” given in Fig. 9 of Narain et al. [4] needs to be replaced by a flat or weakly concave shape for this bowl. Therefore, the existence of steady solutions is at the mercy of other factors such as whether or not an “attracting” steady solution exists and, if it does, whether or not downstream conditions in the experimental set up are conducive to the attainment of this “attracting” solution and associated exit condition. As a matter of semantics, it should be noted that, for category II flows, in Phan et al. [6], existence of an “attractor” leading to a long-term steady exit condition was termed differently – it was called a longterm “one-way” or “parabolic” behavior. Similarly, non-existence of an “attractor” (typically an indicator of flows that lie outside the annular regime and are more complex in the sense that they exhibit certain degrees of randomness or indeterminacy) was termed differently in [6] – it was called long-term “two-way” or “elliptic” behavior. Category III (Complete condensation involving special specified conditions at the inlet and the exit) Though, technically, this is a special subcategory of a general unsteady and/or steady category I flows, it is listed separately because it typically involves and often used specialized experimental set-up and hardware facilities. This class of condensing flows has been extensively investigated in the experimental literature ([12]-[19]) for steady or oscillatory flows. • In this case there is a constant pressure reservoir, with a stagnation pressure pTank-in, that feeds the ⋅ vapor flow (at inlet mass flow rate Min (t) , pressure pin, temperature TV-in, and density V-in) into the test section through an inlet valve (with valve coefficient ki). This requires the inlet pressure, pin, to satisfy at any time t: M (t)2 pin ≅ pTank-in − ki in (1) V-in • Also, there is a constant pressure exit tank, with a stagnation pressure pTank-exit, to which the condensate flows through an exit valve of valve coefficient ke. The exit valve handles an all liquid flow because this case is only for complete condensation flows. At the exit of the condenser, the 9 liquid flow rate is Mexit at any time, t, and the liquid density is section exit pressure, pexit, satisfy at any time t: Mexit (t)2 pexit ≅ pTank −exit + k e L-exit. L − exit • Prescribed or known steady wall temperatures Tw( ) < Tsat(pin) for all condensation occurs. This requires that the test (2) -locations over which film The experimental and/or modeling analysis papers of Wedekind et al. ([12], [15]), Bhatt et al. ([13][14], [16]), Kobus et al. ([19]), Liao et al. ([26], [27]), etc. focus on category III flows for a horizontal condenser. Our theory and experiments also suggest what they find, namely, unless pTank-in and pTank-exit are sufficiently compatible with system parameters (valve coefficients ki and ke) for a desired steady ⋅ value of M in , an oscillatory flow may result with unsteady and oscillatory inlet and exit pressures (viz. pin and pexit). The experiments and computations in this paper, however, focus only on steady category I and category II partial condensation flows. As a result, the flow transients and system instabilities reported in this paper, as far as flows within the test-section are concerned, are necessarily of different origin. However, at a system level, the experimentally observed flow oscillations’ relationship to the better known ([14], [16]) results for category III flows in the downstream auxiliary condenser is also discussed here. The experimental runs reported here largely involve laminar condensate and turbulent vapor situations with possible vapor compressibility effects for some of the category I flows. Despite this, both qualitative and quantitative comparisons with simulation results based on the laminar-vapor/laminarcondensate methodology given in [4]-[6] are possible for a feasible subset (within the boundaries for steady annular flows) of experimental runs. This comparison is presented here and is possible because turbulent vapor often laminarizes in the vicinity of laminar condensate as the condensate is slow and remains laminar approximately up to Re 1800 (see film Reynolds number, Re , definied in Phan and Narain [28]). Also, for gravity driven condensate cases considered here, the existence of turbulent vapor zones in the core and entrance zone of the condenser has only minor second order impact on most of the flow variables with first order effects limited to pressure variations in the condenser. The far field vapor turbulence often tends not to be a significant player because the overall flow features (local and average heat transfer coefficients) are dominated by interfacial mass and heat transfer rates, which are governed by the typically laminar nature of the gravity driven condensate flow and the associated laminar nature of vapor flow in the vicinity of the interface. Because of the above, all experimental runs reported here for partial (or incomplete) condensation cases involving laminar condensate show a very good qualitative agreement with the simulations as far as the existence of multiple steady solutions for multiple steady exit conditions (category I) and a “natural” steady solution for the unspecified exit condition cases (category II) are concerned. The agreement with simulations, with regard to exit vapor quality and general consistency with overall heat transfer rates, are also quantitatively very good for category II experimental runs that fit the annular flow assumption for the simulations. The experiments reported here involve a single pure working fluid (viz. FC-72 by 3M Corp.) and focus on inlet mass flow rates that correspond to inlet vapor Reynolds numbers in the range of 10,000 40,000 and vapor to wall temperature differences of 3 - 60oC (i.e. 0 Ja 0.4). 10 Furthermore, computational results for external condensaing flows that are obtained from this simulation tool have proven record (see Phan and Narain [28] and Kulkarni et al. [2]–[3]) of making qualititively and quantitiavely correct predictions for steady flows. The steady flow predictions are in excellent aggrement with the classical solutions of Nusselt [21] and Koh [29]. Additionally, the unsteady results obtained from this computational tool are also able to accurately predict wave phenomena and their effects for these benchmark classical problems. Hitherto nonexistent accounting of exit condition categories (and exit condition parameter “ϕ” introduced later in this paper for internal condensing flows, as discussed in section 6), is perhaps one of the reasons for the large uncertainities and deficiencies noted by Palen et al. [30] with regard to the poor usefulness of quantitative information available from existing correlations for heat transfer coefficients. Fig. 3: For a channel (h = 0.004 m) in 0g environment and flow of R113 at uniform inlet speed U = 0.6 m/s and temperature difference ∆ T = 5°C, the figure shows non-dimensional film thickness at different non-dimensional times (see [6]). A zone with strong steady attractor is seen in the initial part of the channel and the zone after x* > 23 exhibits weak unsteady attraction to a quasi-steady solution. Instead of the strongly attracting solutions for gravity driven category II flows in a vertical channel (of the type shown in Fig. 2), one obtains significantly weaker attracting solution for horizontal channel or 0g channel flow situations and this is exemplified by Fig. 3 where an attractor exists only for 0 x x* 23 with the attractor’s strength diminishing with increasing distance from the inlet. For x* x x** 45 in Fig. 3, the “attractor” can be seen to have morphed into a rough “attracting zone” that is sufficiently weak and wavy and thickens with increasing x (no initial disturbance were imposed – the simulations are showing amplified computational disturbances). For x >> x** 45 locations that are not shown in Fig. 3, the “attracting zone” is non-existent and true “elliptic” nature of the problem manifests 11 itself for unspecified exit condition category II flows - making the flow physically and computationally indeterminate. Furthermore, it is found that x* increases/decreases with increasing/decreasing inlet mass flow rate (per unit depth of the channel) M in = ρ2Uh and, for x* 0, there is a positive non-zero lower threshold of mass flow rate – termed M in − cr - for which no steady flow exists over any length of the channel. Therefore, if the category II flow is one of partial condensation and the length of the channel is xe, then we have the following possibilities: the flow has an “attractor” if xe < x*, it has a weak “attractor” over x* xe x** if xe is in this range, and the flow has no solution possible under category II conditions if xe >> x**. In other words, fully condensing steady flows of this type (stratified/annular) in micro-gravity or horizontal channel is an impossibility for category II flows. However, as shown later, a fully condensing steady flow under category I or III flow conditions with suitably prescribed inlet pressure, exit pressure, and cooling method (i.e. condensing surface temperature Tw( )) is possible and can be arranged (this allows for non-annular plug-slug flow regime near the point of full condensation) for zero gravity or shear driven horizontal situations. If the exit pressure chosen for realization of category I flow is not compatible with inlet and other conditions, oscillatory flow may result. The above results also make physical sense. The attractor for gravity driven cases under category II flows arise because of the irrelevance of vapor motion as the condensate is driven by gravity and the vapor motion is eventually (near the point of complete condensation) determined by a liquid flow which is allowed to flow and get whatever pressure it seeks. However since the entire flow morphology can easily be changed by a “non-natural” externally imposed prescription of exit-pressure under category I conditions, the category II attainment remains sensitive. For shear driven horizontal or micro-gravity flows, existence of an annular/stratified “attractor” of the type being sought here requires sufficiently fast vapor flows M in > M in − cr that can provide sufficient shear and determine pressure fields over a certain length x* of the duct. However, the definition/existence of this attractor is in jeopardy if the vapor motion is not sufficiently well defined; this which happens, for unspecified exit pressures, after a certain distance x*. Fig. 4a: The figure shows the non-dimensional film thickness and the u versus y velocity profile (at x = 25) for steady solution of the typical Nusselt problem (see [28]) with ∆ T = 5°C. 12 Fig. 4b: For a forced flow with U = 0.2 m/s and all other conditions remaining the same as in Fig. 4a, this figure shows the non-dimensional film thickness and the u velocity profile (at x = 25) obtained from a steady solution of the problem. It can be seen that the film thickness values do not change, as compared to the those in Fig. 4a, but u velocity profiles and associated shear values are different for the two cases. Fig. 5: For a horizontal condensing flow over a flat plate (see [3]) with U = 1.7 m/s and ∆ T = 5°C, this figure shows non-dimensional film thickness at different non-dimensional times given by the unsteady solution of the problem. An initial guess given at time τ = 0 is seen to get attracted to the long term steady solution at different rates. This rate decreases along the length of the plate. The markings demarcate the zones that converged into steady solution at different times. 13 In fact the strong versus weak attractor situations depicted, respectively, in Figs. 2 and 3 for vertical and 0g channel flows is quite typical and are also present, respectively, for their external flow counterparts, namely the Nusselt problem (see [28]) and the Koh problem (see [2]-[3]) as well. Note, however, both the Nusselt [21] and the Koh [29] problems have well defined uniform pressure values at the inlet, far field, and at the finite exit location used to model the infinite plate. Therefore, in a sense, the solution of these problems, particularly the Koh problem [29] for which vapor motion is important, should behave more like a category I than a category II flow problem. The steady solution shown in Figs. 4a and 4b are for Nusselt problem and Nusselt problem in the presence of forced convection respectively. The depicted steady interface locations shown in Figs. 4a and 4b are strong “attractors” as they have also been obtained, though not shown here, in the limit of τ solutions of the unsteady problem with an initial condition different than the long term steady solution itself. As seen from the comparison of Figs. 4a and 4b, for the case considered, the film thickness does not change in the presence of forced convection as the vapor motion remains unimportant. This is because, for both cases, gravity is the dominant force in driving the condensate and the steady solutions shown and generally obtained (see [28]) satisfy the well known analytical results (see [21] and [28]) for film thickness in the presence or absence of waves. In comparison, the representative 0g solutions (obtained by solving the unsteady equations in [2] – [3] for the Koh problem [29]) shown in Fig. 5 are possible only if U is above a certain lower threshold (termed U |L-cr in [3]). Furthermore the solution in Fig. 5 exhibits the presence of an “attractor” whose attracting strength decreases with increasing x. The existence of a threshold inlet speed is similar to what is needed for the channel flow in Fig. 3 – but the Koh problem [29], being a category I external flow problem has an “attractor” which theoretically exists at all x locations (i.e. there is no x* as in Fig. 3). Despite this, in Kulkarni et al. [2]-[3], it is shown that the weak attractor at downstream locations becomes more noise-sensitive and amplifies ever-present minuscule noise (condensing surface vibrations, etc.) leading to persistent and large amplitude waves around Rex|wavy-cr 5 x 105. Further downstream of this, the condensate is expected to become turbulent. 2. EXPERIMENTAL FACILITY The condenser section, which is of the type shown in Figure 1a, is typically a part of a closed flow loop. The flow loop, which maintains a steady inlet pressure (pin) and mass flow rate ( M in ) at the inlet, while maintaining a prescribed steady (and nearly uniform) condensing surface temperature, may be designed to provide different categories of exit conditions. Exit condition specifications for category I and category II partial condensing flows defined earlier in section 1 are realized through flow arrangements indicated in Fig. 6 and Fig. 7 respectively. A 0-500 W evaporator/boiler in Figs. 6-7 is used to evaporate the working fluid (FC-72). The vapor mass flow rate out of the evaporator, Min , is fed into the test section. This mass flow rate is measured by a Coriolis flow meter F1 and, during transients, this value can be controlled by the pneumatically actuated control-valve V1 (shown connected to F1 in Figs. 6-7). Under steady conditions though, the value of Min gets approximately fixed by the net steady electrical heating rate for the evaporator. This is due to the restriction imposed by the evaporator energy balance, viz., M in ≈ QB / h fg (pB ) . Here QB is the net heat rate into the evaporator, pB is the steady evaporator pressure, TB Tsat(pB) is the steady evaporator temperature (which is nearly equal to the saturation temperature of the fluid at pressure pB), and hfg is the heat of vaporization at the liquid/vapor surface pressure pB in the evaporator. Towards reduction in start-up time to steady state in the evaporator, the liquid flowing in the evaporator is warmed up, 14 (between points P' and B' in Figs. 6 – 7) so its temperature is nearly equal to the evaporator temperature TB Tsat(pB). Fig. 6: The schematic of the flow loop for achieving specified exit condition category I flows for partial condensation cases. The test-section is a 0.8 m long, vertical, stainless steel (316 SS) tube with 6.6 mm inner diameter, D, and 12.7 mm outer diameter. At the entrance of the test-section, the inlet vapor temperature is denoted as TV-in, the inlet pressure is denoted as pin, and the inlet vapor is kept slightly superheated (210oC superheat obtained by heating a relevant portion of connecting tubes by a rope heater). A suitable thermocouple and an absolute pressure transducer respectively measure the temperature, TV-in, and pressure, pin, of the vapor at the inlet. The dynamic view from an axial boroscope, mounted at the top of the test-section shown in Fig. 8, is used to visualize and ascertain the nature of the flow in the first half of the test-section. However, because of sharpness and contrast improvements that are needed for better quality images, snapshots and video clips of the flows are not included in this paper. They are, however, expected to be available in the near future. We are currently able to use these views to ascertain whether or not annular film condensation begins near the indicated “start of condensation” point in Fig. 8 and, also, to ascertain (and then to ensure) dryness of the vapor up to the test-section inlet. The test-section (see Fig. 8) is suitably instrumented with various sensors (thermocouples, pressure transducers, etc.). For future work, it will be possible to obtain local film thickness data through integration of our recently invented, non-intrusive film thickness sensors that utilize the principle of fluorescence and fiber-optic technology (see Ng [31]) and are able to measure the “local” time-varying 15 thickness of dynamic liquid films. The technique used for mounting all the sensors in Fig. 8 is described in detail in [32] - [34]. Fig. 7: The schematic of the flow loop for achieving unspecified exit condition category II flows for partial or full condensation cases. The test-section in Fig. 8b (not shown to size relative to the outer tube in Fig. 8a) is centrally aligned in the hollow space of a larger diameter stainless steel (314 SS) tube. This outer tube has an inner diameter of 23.62 mm and an outer diameter of 25.40 mm. The test-section tube is cooled by the flow of cold water in the annulus formed by the outer surface of the test-section tube and the inner surface of the outer tube. As shown in Figs. 6-7, the flow of coolant water is arranged by a separate closed loop consisting of the shell-side of the shell-in-tube heat exchanger (flow is on shell-side) and a pump. A separate loop, not shown in Figs. 6-7, assures secondary coolant (cooler-water) flow at a steady constant temperature and a steady flow rate through the tube-side of the heat exchanger in Figs. 6-7. This loop (see Kurita [33]) replaces the open drain water loop used in the preliminary experiments. This loop has two chillers in series (one for coarse and one for finer control of temperature) and this provides for a good control of the steady value of temperature TC-in (marked in Fig. 8a) at the coolant inlet location in Figs. 6-7. This, in turn, enables repetition of experimental runs regardless of seasonal variations in drain water temperature. Resistance temperature detectors (RTDs) and type-T thermocouples measure temperatures at different locations of the test-section (see Fig. 8) and at other flow loop locations marked by points B, B′ , T ′ , C1, C2, D1, D2, P1′ , P2′ , and P′ in Figs. 6-7. A barometer measures outside atmospheric pressure. 16 Flow meters at locations marked F1 (Coriolis meter that directly measures mass flow rate), F2 (a volume flow rate measuring rotameter), P1 (volume flow rate meter imbedded in pump P1), and P2 (volume flow rate meter imbedded in pump P2) yield mass flow rates through those locations. Absolute pressure transducers measure pressures at test-section inlet (location 1 in Fig. 8), and at locations B and D2 in Figs. 6 - 7. Differential pressure transducers measure pressure differences in the test-section (in Fig. 8a, this is between locations 1 and 9, location 3 and outside atmosphere, and location 6 and outside atmosphere). Two electronically controllable displacement pumps P1 and P2 (see Figs. 6-7) can pump liquid FC-72 at a steady or unsteady specification of volume or mass flow rates. A pneumatically controlled valve V1 is used, as needed, to control mass flow rate through F1. Most of the details of the employed data acquisition system are explained in Narain et al. [34] and Siemionko [32]. Fig. 8: (a) The photograph of condenser test-section. (b) The test-section schematic (diameters in (a) and (b) are not to the same scale). The condensing surface covers the zone 0 10. 17 For convenience, the system in Figs. 6-7 is broken into the following sub-systems. (i) Sub-system is the portion of the flow loop between points P′ and T ′ (this portion contains the flow into the evaporator, the evaporator, the flow meter F1, valve V1, and the tubing leading the flow into the testsection). (ii) Sub-system is the portion of the flow loop between points T ′ and T" (this portion consists of the test-section). (iii) Sub-system is the portion of the flow loop between points T" and P′ (this portion consists of the L/V separator, the two branches of the flow in the liquid line and the auxiliarycondenser line, and the pump or pumps). (iv) Sub-system is the portion that consists of the primary coolant loop shown in Figs. 6-7 and a secondary coolant loop (not shown here but shown in Kurita [33]). The sub-systems defined above are not marked in Figs. 6-7 but the definitions introduced here are necessary for later discussions of the experimental results. Additional details of this experimental facility (rather of a similar earlier version), data-acquisition, and LabView 7.1 based data processing and instrument control strategies are available in Siemionko [32]. 3. EXPERIMENTAL PROCEDURE Here we describe the procedure for investigating partial condensation cases under different exit conditions. Some procedure and results for fully condensing cases are also given in Narain et al. [1]. Note that results from different exit conditions are to be compared for approximately the same inlet mass flow rate M in , inlet pressure pin, and temperature difference T = T sat ( p in ) − Tw , where Tw is the mean condensing surface temperature. Pure vapor may be allowed to enter the test section with a superheat of 2-10oC. The purging process (see [32]) ensures that the vapor flowing in the test section is pure over the duration of the experimental run and that non-condensable air in the flow loop has zero to insignificant presence. Note that vapor Jakob numbers Ja|V ( Cp2 Tsup/hfg(pin)), where Cp2 is the specific heat of the vapor and hfg(pin) is the heat of vaporization at pressure pin) represent the ratio of sensible cooling of vapor to heat of vaporization. Since these numbers are very small (< 1.0 ⋅ 10-5) in comparison to liquid Jakob numbers Ja ( Cp1 ∆ T /hfg(pin)), where Cp1 is the specific heat of the liquid condensate), vapor temperature can be effectively modeled as a steady constant equal to the inlet saturation temperature. The steadiness of Min primarily depends on the constancy of heat supply to the evaporator (which is easily achieved by constant electric heating at a known wattage) and the eventual approximate steadiness of the evaporator pressure pB. Even if the steady value of pB changes somewhat for different start-ups, as long as the corresponding evaporator saturation temperature TB = Tsat(pB) changes negligibly, it is found that the remaining flow loop pressures relative to a single effective boiler pressure are the same for two independent repetitions of the same experimental run (i.e. same Min , ∆ T , Tw , and Ze). This is true because all other pressures, p, are effectively characterized by their pressure differences, p-pB, relative to this pressure pB. Alternatively, a previously obtained steady value of pB can be regained by bringing the new evaporator pressure back to roughly this same value by: (i) use of a newly developed electronically controlled accumulator discussed in section-6, or (ii) by suitably switching the heater on and off under same steady heating rate, or (iii) by gradually venting the evaporator from a higher pressure to the desired earlier value of pB. The last two of the above described processes have been used to successfully assess the repeatability of a few representative experimental runs. With regard to constancy of water temperature TC-in at the coolant inlet for the test-section (see Fig. 8) over time; achieving constancy of water temperature and its flow rate in the secondary coolant loop (not shown here) proved to be sufficient. 18 Specified Exit Conditions (Category I flows) Incomplete or partial condensation flows For investigation of specified conditions (through a steady and specified exit vapor quality) at the exit that involve partial condensation flows through the test-section, the arrangement in Fig. 6 is used. In this arrangement, the liquid at the exit flows out of the test-section at a mass rate of M L , goes through the liquid/vapor (L/V) separator, and is pumped by pump P1 back into the evaporator. Both the pumps P1 and P2 (displacement pumps made by Masterflex) in Fig. 6 allow digital control of flow rates. The vapor at the exit flows out of the test-section at a mass flow rate of MV and is measured through a volume and mass flow rate measuring rotameter F2. This vapor then flows through an auxiliary condenser where the vapor is completely condensed into liquid, goes through the pump P2, and then, on its way to the evaporator, merges near point P′ (see Fig. 6) with the liquid flowing out of pump P1. The control strategy, to achieve a specified steady flow with a prescribed exit vapor quality ( M V / Min Ze) for a given inlet and wall conditions, is to initially hold valve V1 open at a fixed level of opening while ensuring (as described in the first paragraph of this section) desired steady values of Min , . Then the exit vapor mass rate M V through pump P2 (or rotameter F2) is held fixed at a value less than the inlet mass rate Min while exit liquid mass flow rate M is varied through pump P1 at a pB, and ∆T L value given by the tracking equation: M L |P1 = M in − M V |Rotameter . As the flow through the evaporator becomes steady, Min becomes steady, and, at that time, we may or may not need to hold this value actively fixed with the help of controllable valve V1. At this stage, active control of valve V1 does not achieve much except that it eliminates some unwanted minuscule drifts in the inlet mass rates. For a given set of inlet ( Min , pin, TV-in) and wall ( Tw ) conditions, different specified steady states are achieved by the above strategy for different values of M L . Examples of experimentally achieved partial condensation flows under specified exit conditions (category I flows) are given and discussed in the next section. Unspecified Exit Condition Cases (Category II flows) “Natural” Partial Condensation For obtaining/investigating existence of a “long term,” steady, “natural” exit condition for category II flows (under unspecified exit conditions) with all other conditions being kept the same as in a corresponding specified exit condition case in category I, the flow is required to go through the testsection and onwards under the arrangement shown in Fig. 7. Note that this arrangement has a single displacement pump as opposed to the two displacement pumps used in the arrangement of Fig. 6. The approach is to hold values of M in , pB, and ∆ T nearly the same as in one of the specified category I cases while the pump P1 in Fig. 7 is controlled such that the mass flow rate through it tracks the equation: M L| P1 = M . If start-up and other conditions (such as the range of pressures allowed in the L/V separator) allow a steady state flow, with gravity driven condensate, is attained in which, at the exit of the test-section, the inlet vapor mass flow rate is split, by a natural selection process, into a liquid condensate flow rate ML |Na and a vapor flow rate M V | Na . Clearly these values satisfy the equation: M in = M L | N a + M V | N a . Examples of experimentally achieved partial condensation flows under unspecified exit conditions (Category II flows) are given and discussed in the next section. in 19 4. EXPERIMENTAL RESULTS, DISCUSSIONS, AND COMPARISONS WITH SIMULATIONS The column headers in Tables 1-2 indicate accuracies of the values of key measured variables obtained through flow loops’ instruments and sensors. Overall accuracy bounds for the reported calculated variables (such as q′′ , h , etc.) are also shown. The non-dimensional numbers Rein, xe = L/D = 106, Ja, ρ2/ρ1, and µ2/µ1 in Tables 1-2 define the flows and they are defined in [4]-[6]. In Tables 1-2, the heat flow rate Q out and associated average heat transfer coefficients ( h ) are obtained through the relation: Q out ≈ M L h fg = h ( DL) ∆T . The inlet vapor mass-flux G in Tables 1-2 is defined as 4 M in / (π⋅D2). All (except the Coriolis meter F1) of the instruments’ accuracies for measured variables were established after their in-house calibrations with the help of suitable and reliable reference instruments of known resolution and appropriate reference physical conditions (temperature, flow rate, pressure, etc.). The accuracy of the Coriolis meter was established by the vendor support staff at the time of its installation. The error estimates for the calculated variables reported in Tables 1-2 were obtained by well-known standard procedures (see, e.g., eqs. (3.27)-(3.28) in Parratt [35]). The accuracies of individual calculated variables in a column were taken into account to report maximum values of the errors in the column headers of Tables 1-2. All the individual values of errors were either less than or equal to these reported error values. The error definitions, associated error analyses and calibration accuracies can be found in Narain et al. [34]. Table 1: Experimentally measured data and some key calculated and computed variables for steady states achieved for category II (unspecified exit condition) partial condensation flows. Run M in MV Ze Ze EXP COMP p pD2 2/ 1 2/ 1 G Re Ja Pr1 q′′ Tw Tsat ∆T pin px6 pT" (K) (K) (K) (kPa) (kPa) (kPa) (kPa) (kPa) ±1 ±0.15 ±1 ±0.6 ±0.6 ±0.7 ±0.05 ±2 ±0.0001 ±0.0001 ±1.5 ±900 ±0.02 ±0.02 ±5 ±800 ±80 Qout h No. (kg/m2 s) (g/s) ±0.05 ±0.04 ±0.04 1 1.44 0.48 0.33 0.33 320 331.49 11 107.3 109.3 106.5 -0.82 105 0.0085 0.0243 42.1 23900 0.14 9.61 81 5200 453 2 1.76 1.08 0.62 0.57 317 325.23 8 87.0 86.4 88.0 -0.36 85 0.0070 0.0223 51.5 29700 0.10 10.11 58 3700 450 3 1.54 0.69 0.44 0.36 323 335.55 12 122.6 116.6 116.1 -0.19 114 0.0097 0.0260 45.0 25300 0.15 9.21 71 4500 387 4 1.29 0.49 0.38 0.38 320 329.64 10 101.0 102.4 100.8 -2.09 98 0.0081 0.0230 37.7 21500 0.11 9.69 68 4300 476 5 1.70 0.83 0.51 0.52 324 332.55 9 111.0 112.5 111.0 -0.93 109 0.0089 0.0252 49.6 28100 0.11 9.37 70 4400 508 6 1.17 0.47 0.40 0.39 320 332.64 13 111.3 NA 111.3 -0.12 109 0.0088 0.0246 34.2 19325 0.16 9.55 59 3700 298 7 1.31 0.49 0.37 0.37 321 330.85 10 105.0 106.8 104.5 -2.06 102 0.0084 0.0244 38.3 21700 0.12 9.58 69 4400 462 8* 1.93 1.39 0.72 0.72 322 325.55 4 84.6 NA NA 0.01 NA 0.0071 0.0231 56.4 32500 0.05 9.85 47 3000 742 9 1.59 1.11 0.69 0.63 328 334.25 6 113.0 NA NA NA NA 0.0094 0.0265 46.5 26200 0.08 9.07 40 2500 418 10 2.12 1.37 0.64 0.64 320 327.85 8 91.4 NA NA -0.10 NA 0.0076 0.0234 62.0 35500 0.10 9.83 64 4100 503 11 1.30 0.45 0.35 0.38 321 329.29 8 99.8 100.2 99.7 -1.62 97 0.0080 0.0240 38.0 21700 0.10 9.70 72 4600 537 20 (J/s) (W/m2) (W/m2 K) (g/s) Table 2: Experimentally measured data and some key calculated variables for steady states achieved for category I (specified exit condition) partial condensation flows. Run M in M V Ze-exp Tw Tsat T pin px6 pT" (K) (K) p pD2 (K) (kPa) (kPa) (kPa) (kPa) (kPa) 2/ 1 2/ 1 G Re Ja Pr1 Qout (kg/m2 s) h (W/m2) (W/m2 K) (g/s) (g/s) ±0.05 ±0.04 ±0.04 ±1 ±0.15 ±1 ±0.6 ±0.6 ±0.7 ±0.05 ±2 ±0.0001 ±0.0001 ±1.5 ±900 ±0.02 ±0.02 ±5 ±800 ±80 1 1.44 0.52 0.36 320 331.91 12 108.8 110.9 108.0 -1.82 106 0.0087 0.0244 42.1 23800 0.15 9.59 78 4900 416 2 1.31 0.50 0.38 320 329.16 9 99.4 100.3 99.5 -0.43 98 0.0080 0.0238 38.4 21900 0.11 9.72 69 4400 505 3 1.34 0.59 0.44 320 328.55 9 97.4 98.4 97.3 -0.32 96 0.0078 0.0236 39.1 22400 0.10 9.76 64 4100 491 4* 1.80 0.64 0.36 323 330.53 8 103.9 105.9 104.4 -0.59 103 0.0083 0.0246 52.6 29900 0.09 9.50 97 6200 851 5* 1.92 1.00 0.52 323 329.64 7 101.0 103.3 101.7 -0.40 100 0.0081 0.0244 56.1 32000 0.08 9.55 78 4900 771 6 1.20 0.00 0.00 320 332.23 12 110.5 113.2 110.5 -0.80 108 0.0088 0.0245 35.0 19800 0.16 9.57 101 6400 514 7 1.22 0.44 0.35 320 331.71 12 108.0 110.6 108.0 -0.62 106 0.0086 0.0243 36.6 20200 0.15 9.62 66 4200 351 8* 1.93 1.15 0.60 322 325.21 3 87.15 NA NA -1.30 NA 0.0071 0.0231 56.4 32600 0.04 9.87 67 4300 1174 9 1.61 1.14 0.70 322 332.57 11 111.4 104.6 107.0 -0.87 103 0.0089 0.0250 47.1 26630 0.13 9.46 35 2200 212 10 1.67 1.37 0.82 322 331.45 10 105.8 98.6 100.8 -0.74 97 0.0085 0.0245 49.0 27800 0.12 9.57 16 1000 108 11 1.81 1.02 0.56 317 331.48 14 106.8 102.1 NA -0.58 99 0.0085 0.0237 52.9 30000 0.18 9.80 54 3400 236 12 1.66 0.60 0.36 318 333.15 15 113.5 108.6 NA -1.46 106 0.0090 0.0243 48.5 27400 0.20 9.66 88 5600 363 13 1.71 0.49 0.28 316 331.14 15 108.0 107.9 108.1 -2.10 105 0.0085 0.0237 50.1 28400 0.19 9.81 103 6500 441 14 1.30 0.65 0.50 320 326.34 6 90.5 92.1 90.5 -1.15 88 0.0073 0.0230 38.0 21800 0.08 9.90 56 3500 545 15 1.29 0.55 0.43 321 331.63 11 107.7 113.2 110.1 -1.60 107 0.0086 0.0245 37.7 21400 0.14 9.57 62 3900 366 16 1.31 0.76 0.58 321 331.07 11 105.8 109.1 105.8 -0.95 104 0.0084 0.0243 38.3 21700 0.13 9.62 46 2900 279 17 1.14 0.88 0.77 319 326.73 8 91.6 92.8 91.9 -0.14 90.4 0.0074 0.0230 33.5 19300 0.09 9.92 22 1400 186 18 1.39 0.64 0.46 318 329.01 11 98.9 101.3 98.4 -0.65 97.1 0.0079 0.0234 40.6 23200 0.14 9.85 64 4100 377 No. (J/s) q′′ The experimental runs reported in the next section were taken after ensuring that: (i) representative runs were repeatable, (ii) the mass flow rates for partial condensation cases added up to satisfy mass . balance, (iii) overall energy balance for the test section was satisfied i.e. M in hfg . M w Cpw Tw , where, Tw is the rise in the water temperature in the annulus surrounding the test-section, and (iv) various data were reasonable (based on simulation estimates) and consistent with one another. The experimentally obtained partial condensation cases in category II (unspecified exit) are listed in Table 1 with all the essential details including exit vapor quality Ze (fourth column) and its value obtained from simulations (fifth column). Note that the fourth and fifth columns are in good agreement with one another. The corresponding partial condensation flow cases under category I (specified exit condition cases) are listed in Table 2. The discussion for these partial condensation flow results for exit condition categories I and II are given next. 21 Fig. 9a: Time history depiction of Min , MV and ∆ T values for multiple steady states of partial condensation cases viz. Natural-1 (run 1 from Table 1), Specified-1 (run 1 from Table 2), Specified2 (run 18 from Table 2), and Specified-1 Approx. Test Section (Sub-System ) Pressures px6 Px6 pin Pin px9 Px9 Pp 120 5 116 4 Natural-1 Specified-1 Inlet Pressure Specified-1 Approx. Specified-2 3 108 2 Inlet Disturbance 104 1 Inlet Disturbance Exit Pressure 100 0 . P ( kPA) Pressures: pin, px6, px9 (kPa) 112 -1 96 P 92 -2 Transient D 88 -3 Transient D 84 -4 76 min. 20 min. 80 0t 1 20 20 minutes 40 60 t2 80 30 min. t3 100 20 min. 120 t4 -5 140 Time Fig. 9b: Time history of pressures (along the test section) and p values (across the test-section) for the cases shown in Fig. 9a. 22 Test Section (Sub-System ) Temperatures Tx1 Tx4 Tx9 Saturation Temperature 60 Inlet Disturbance 58 Inlet Disturbance Transient D 56 54 Temperatures (°C) Transient D 52 Wall Temperatures Specified-1 Natural-1 Specified-1 Approx. Specified-2 50 . 48 46 44 42 76 min. 20 min. 30 min. 20 min. 40 0t 1 20 20 minutes 40 60 t2 80 t3 100 Time 120 t4 140 Fig. 9c: Time history of temperature values along the test section (sub-system ) and Tsat (pin) for the cases shown in Fig. 9a. Partial Condensation Flows Specified Exit Condition (Category I) and Unspecified Exit Condition (Category II) Cases The results over time interval t4 t (min) t4 + 20, termed “Specified-1 Approx.” show the experiment’s ability to approximately repeat the data for a case that is approximately the same as the one for t2 t t2 + 20 termed Specified-1. Following the method described in section 3, Figs. 9a-9b also show, over the time interval t1 t (min) t1 + 76, the attainment of a corresponding “natural” steady exit condition and associated steady flow variables for an unspecified exit condition (category II) case. This case corresponds to run 1 in Table 1. The Natural-1, Specified-1 and Specified-2 steady states (in Fig 9a – 9c) have the same values of M in 1.44 ± 0.05 g/s and ∆ T 11 ± 1oC but different values of M L and MV that satisfy M L |1 + M V |1 = M L |2 + M V |2 = M in . The differences between the Specified-1 and Specified-2 cases are: (i) they have different heat transfer rates (since, energy balance gives: Q out ≈ M Lh fg ), the two cases respectively have approximate heat transfer rates of 78 ± 4 W and 64 ± 4 W and average heat transfer coefficients of 416 ± 40 W/m2-K and 377 ± 40 W/m2-K, and (ii) different hydrodynamics – the signature of which is clear through corresponding computational simulations and, also, through the difference between experimentally obtained mean values of p for the two cases (they are, in Table 2, respectively, -1.82 kPa and -0.65 kPa). Furthermore, specified (category I) and unspecified (category II) flows have different dynamic responses to a disturbance (in Figs. 9a-9c, a disturbance was induced by momentarily shutting or decreasing the opening in the valve V1 shown in Figs. 6-7). The difference in dynamic response is seen by comparing Specified-2 and Natural-1 cases for transients’ decay time Dτ 23 associated with exit vapor flow rate M V in Fig. 9a or Dτ associated with p in Fig. 9b. In Fig. 9a, the rapid shutting or closing of valve V1 caused the indicated responses in Min time history. For the Natural1 case, this response is as shown in Fig. 9a but a more rapid response for Specified-2 case is not captured by the resolution of the figure and it is indicated by a dotted line. With regard to dynamic responses to a disturbance – it is clear that Specified-2 case of Fig. 9a (though it is farther from a “natural” case) is more stable than the case for Natural-1 because its transients decay time Dτ is much shorter. In other words, “natural” steady states for unspecified exit conditions (category II) are generally more noisesensitive because the exit, in category II cases is not as isolated from flow variations further downstream of it as is the exit for specified exit condition cases (category I) - and this causes additional lingering impact of noise arising from the flow variables in the exit zone. Time histories of pressure and temperature values in sub-system , , and for the cases in Figs. 9a-9c are not shown here for brevity but are available in Figs. 7d-7e of Narain et al. [34]. The cases shown in Fig. 9a-9c are representative runs taken from a set of partial condensation runs for specified (category I) and unspecified (category II) exit condition cases in Tables 1-2. The data matrix associated with these partial condensation category I and category II cases is best represented by Fig. 10. The test matrix for all partial condensation (including both the categories I and II) cases is limited by the system limits and flow regime boundaries indicated on the plane marked by inlet mass flow rate M in and temperature difference ∆ T values. Figure 10 shows all the partial condensation cases plotted on the two dimensional plane formed by M in and ∆ T . These parameters were found to be the key variables controlling the dynamics of the condensing flows in the test section. Test Matrix for Category I and Category II Partially Condensing Flows 15 The region encompassing data matrix 13 Zone of Full Condensation 11 Curve - C Curve - D with computationally obtained two points on it __ T ,°C 9 Curve - A with experimentally obtained two points on it Oscillatory or unsteady flow 7 (System induced) 5 Zone of wispy Annular Flow 3 1 0.69 Curve - B with experimentally obtained three points on it 0.89 1.09 1.29 . 1.49 Dry patches/No annular Flow 1.69 1.89 2.09 2.29 Min, g/s Fig. 10: Two-dimensional test data matrix for category II (unspecified exit condition) partial condensation cases’ points and different bounding curves represented on Min - ∆ T plane. 24 . M in , Tsat , pin Fig. 11: The schematic of full condensation showing “point of full condensation” and its downstream region with zero interfacial mass and heat transfer. The typical values for lower and upper limits for inlet mass flow rate were found to be 1 g/s and 2 g/s respectively and that for the ∆ T were recorded to be 2 °C and 12 °C respectively. The interior shaded zone in Fig. 10 represent M in and ∆ T values for which steady flows were attained for both specified (category -I) and unspecified (category II) exit condition cases. The bounding curve-B in Fig. 10 indicates lower threshold of ∆ T such that steady condensing flows attained below that curve (see points marked in Fig. 10) were drop wise patchy – i.e. not annular – on the condensing surface near the inlet. Below this curve, the condensation – as observed from the inlet boroscope – indicates that the flow is no more film annular near the point of onset of condensation as there are wet and dry patches associated with drop-wise condensation. This happens because ∆ T value is below a lower threshold. The bounding curve-B is partly experimental and curve-C on the right in Fig. 10 is, at present, entirely schematic (i.e. not fully explored by experiments). Curve-C represents expected transition to wispyannular flows (see Fig. 10.3 in Carey [36]) at very high Min at any ∆ T . The dotted curve-A on the left bottom has been experimentally noticed. This is a system-instability curve and does not represent a flow regime boundary for the test-section, as it is a result of the exit pressure oscillations or unsteadiness in test section imposed by oscillatory or other plug/slug instabilities occurring elsewhere in the system (in this case, in the auxiliary condenser downstream of the test section). An example of such an instability case is discussed in section 5. The bounding curve in the upper left corner of Fig. 10 is marked as curveD. This curve represents transition from partial condensation to full condensation. If M in is reduced and ∆ T is increased further, computations show that the left side of curve-D represent the zone for which the entire vapor coming in condenses inside the test section (i.e., for category II flows, fc in the schematic 25 of Fig. 11 starts satisfying fc L on the left side of curve-D as opposed to fc > L on the right side of curve-D). Note that Fig. 11 suggests, in accord with computations as well as remarks of Rabas and Arman [11], that this point of full condensation does not necessarily imply an all liquid phase for > > fc is such that average vapor mass flow rate MV is zero and so fc. It simply means that the zone are the interfacial mass and heat transfer rates at these locations. For a few data points in Fig. 10, the rotameter F2 data was corrupted by the float’s occasional stickiness to the rotameter walls. These cases are marked by unfilled circles in Fig. 10 and all the rest of the good cases (also based on comparisons with computational simulations) are marked by dark filled circles. These dark filled circles representing good partial condensation cases in Fig. 10 are actually the projections on the Min - ∆T plane of the points reported in the three dimensional data matrix which has M in , ∆ T and Ze ( MV / Min ) as three axes. This three-dimensional data matrix is not shown here for brevity but is shown in Fig. 8b of Narain et al. [34] where the figure is able to depict all the cases of category II (unspecified exit) as well as category I (specified exit) partial condensing flows. Each point in Fig. 10 represents, for given values of M in and ∆ T , a set of data consisting both category I cases and its associated unique, “natural” category II case. There are, however, as seen in Fig. 8b of Narain et al. [34], some data sets representing only category I or category II cases. For each data set in Fig. 10 consisting of projections of category I and associated category II cases, category I flows were found to be more robust and stable as compared to their associated category II counterparts. Comparisons with Relevant Computational Results (Partial Condensation) Figure 12 shows the computationally obtained (employing the tools reported in [6]) details of local film thickness and heat flux variations for the specified and unspecified “natural” cases (under noise-free conditions) marked as Specified-2 (run no. 18 in Table 2) and Natural-1 (run no. 1 in Table 1) in Figs. ⋅ 9a-9c. The variation of vapor quality Z ( ⋅ M v / Min at any LC . x) along the test section can be easily obtained from the graphical results in Fig. 12 from the relation Z( ) ≡ 1 − {( ⋅ D)/(M in ⋅ h fg )} q′′w d . 0 The exit vapor quality for Specified-2 case was greater than that of the associated Natural-1 case. As . a result, higher amount of vapor condenses for Natural-1 case and this makes heat transfer rate Qout for natural case to be on the higher side (see Table 1 and 2 for details). For all other conditions remaining the same, as observed from the computational results in Fig. 12, this makes Natural-1 case’s liquid film thickness to be lower and wall heat flux to be higher than the values for Specified-2 case. Such details of representative local variations in film thickness and heat-flux are very important and should be more extensively synthesized with experimental results before heat-transfer correlations are developed for suitable categories and sub-categories of internal condensing flows. However, reliable experimental information on “local” spatial variations of these quantities is not expected until forthcoming incorporation of film thickness sensors and heat-flux sensors in these experiments. Observe that the computationally obtained prediction of “natural” exit vapor quality Ze|Na Comp ( 0.33) for category II flow in Fig. 9a-9c is in a very good agreement with the experimentally obtained Ze|Na Expt ( 0.33) value (see Table 1). In fact, a very good agreement between Ze|Na Comp and Ze|Na Expt values was found for all category II cases in Fig. 10 and this is clear from their numerical values in Table-1. Note that good agreement between experimental and theoretical Ze values have also been obtained and 26 reported (see [5]) for shear driven condensing flow in a channel (category II experiments of Lu and Suryanarayana [37]) of gap height h and length L are such that L/h xe < x*, where x* has the same meaning as in Fig. 3. Fig. 12: For the “Natural-1” and “Specified-2” flow cases in Figs 9a-9c, this figure shows the computationally obtained representative film thickness and wall heat-flux variations along the test section. The film thickness and heat flux values shown have been obtained for smooth interface conditions. In reality, they are modulated by waves in presence of noise (see [6]). The values of pressure drop p (given by pin - pexit) obtained from simulations for all the category II partial condensation cases were negative and below 50 Pa, indicating pexit was greater than pin for all the condensation cases in given M range. Thus the pressure rises across the test-section. The pressure rise is needed to decelerate the vapor, feeding the gravitationally accelerated condensate at the rate needed for condensate motion under unspecified exit pressure condition. This is confirmed by the experimental values of p (see Tables 1 and 2) which are also all negative (except a very few cases). However, as expected, the magnitudes for experimental values of p were found to be greater than those from simulations. The reason behind this is that the simulations assume laminar vapor/laminar liquid flows while, in reality, the vapor Reynolds number are in the higher range (20000-30000); and this makes in 27 vapor flows significantly turbulent in the core (see Tables 1-2), but still laminar near the interface. The turbulence in the vapor core does not affect the mass transfer across the interface by much because condensate motion is gravity driven and the flow of the condensate and near interface vapor is laminar in nature. However, turbulent vapor core significantly increases needed p values for the vapor domain while still keeping its impact on the gravity driven condensate motion negligible. Because of this, the values of vapor quality obtained from the simulation are in good agreement with the experiments but the values of pressure drop p obtained from experiments are higher in magnitudes than those predicted by the assumption of fully laminar vapor flows. However, predicted pressure drop p values do become comparable to experimental values if the predicted interface location is retained and k-ε model (see Narain et al. [34]) is used for the turbulent interior of the vapor while retaining the laminar shear stress estimates at the interface. The modifications in computational procedure needed to account for vapor turbulence are briefly discussed in Narain et al. [34]. () * Q out M in + ∆p Natural $ ∆p∆p==∆∆pp Natural ∆∆p = ∆∆ppff1 $ , %& %' M in ML pP Qin ML ! M L = M in" # " Fig. 13a: Specially designed ideal experimental arrangement. This is a specialized arrangement for a condenser (experiencing complete condensation) in a “thermal system” that transfers heat from a high temperature location to a low temperature location. The specialty of this arrangement is that it can be used to keep exit pressure of the condenser unspecified (by not using the accumulator option for A) or to specify it with precise control (by using the accumulator option for A). It should be noted that the pressure controlling accumulators, whether they are used for the evaporator B or the L/V separator A, accomplish their task by adjusting the amounts of vapor that is exchanged (come in and go out) between the accumulator and the liquid reservoir of interest (A or B). 28 Q out M in p p ≡ p* p1 ≡ p B -pin ML p 2 ≡ p exit − p p p Total System ≡ p1 + p+ p 2 Q in Fig. 13b: Example of an experimental arrangement for which condenser exit conditions are not directly controlled. This system, a variation of the system in Fig.13a, is again for a condenser (experiencing complete condensation) in a “thermal system” that transfers heat from a high temperature location to a low temperature location. This arrangement does not allow for the condensing flow to seek a natural pressure drop as the system decides this pressure drop. This is essentially a specified exit condition flow with the specification being decided by the system (see section 6). Fully Condensing Flows The experimental data and discussions for category II fully condensing flows are not reported here for brevity but can be found in Narain et al. [1]. The experimental investigations of fully condensing category I flows are currently in progress. However the essential ideas for these experiments are discussed here. For fully condensing flow cases the temperature difference ∆ T for a given inlet mass flow rate Min in the experimental set up of Fig. 7 is such that that there is zero mass flow through the depicted auxiliary condenser and, therefore, valve V3 can be considered closed. For the case of unspecified exit pressures (category II flows), the flow loop in Fig. 7 makes available a range of exit pressures for the flow in the test-section by allowing a range of values for temperature TD1 in the L/V separator at point D1 in the flow loop of Fig. 7. As a result, the pressure p = psat (TD1) in the L/V separator can vary over a range because temperature TD1 can take any value between the ambient temperature and the temperature at the exit of the test-section. This special arrangement for fully condensing flows under unspecified exit pressure conditions at fixed boiler pressure pB is best indicated by the simplified flow-loop in Fig. 13a where the L/V separator location is now marked by point A (instead of the marker D1 used in Fig. 7). For the case discussed above, the L/V separator at point A in Fig. 13a is not connected to an active pressure controlling accumulator. In practice, many condensers employ this L/V separator by making it an integral part of the bottom portion (which is not cooled) of 29 the test-section condenser – this is effectively and practically achieved by using an orifice-plate (i.e. a significantly smaller diameter tube than the test-section) at the exit of the test-section. For specified exit pressure fully condensing flow cases (category I), besides the boiler in Fig. 13a, the L/V separator at point A is also connected to an “active” pressure controlling accumulator which holds the pressure fixed at a specified value. For the complete condensation condenser in the system of Fig. 13a, an inadvertent design – which does not consciously keep the exit pressure specified or unspecified – is indicated in Fig. 13b. 5. RESULTS/COMMENTS ON SYSTEM INSTABILITIES AND OSCILLATORY FLOWS While seeking the “natural” exit condition for some unspecified exit condition (category II) partial condensation flows, system instabilities – involving oscillatory flows – of the type shown in Figs. 14a 14b are observed. The origin of these oscillatory flows appears to be the auxiliary condenser, which sees an approximate category III flow for which, in Fig. 6, the L/V separator at point D1 is at approximately fixed stagnation pressure and the L/V separator downstream of the auxiliary condenser at point D3 is at another approximately fixed stagnation pressure. These pressures respectively correspond to pressures pTank-in and pTank-out that appear in the definition given in section-1 for category III flows. Recall that these fully condensing flows in the auxiliary condenser, unlike the ones studied for the test section, are known (as in [14]) to become oscillatory under certain choices of pressures pTank-in and pTank-out. Since complete auxiliary condenser flow data were not obtained (because this component was not the focus of the reported investigations), relating this auxiliary condenser instability to the type of stability boundaries discussed in [14] and [16] is outside the scope of this study. It is clear from Figs. 14a-14b that oscillations in vapor mass flow rate at the exit of the test section impose oscillations on the exit pressure and on the pressure drop ∆p across the test section while the inlet vapor mass flow rates remain relatively unaffected. Figure 14b shows the oscillations in other pressure values and the temperature at the rotameter F2 (which is nearer to the auxiliary condenser). This, along with the known fact (see Fig. 9 in [4]) that there is one to one relation between exit vapor quality and exit pressure, indicate imposition of oscillatory pressures at the exit of the test-section. It suffices here to note that the flow oscillations in the auxiliary condenser can induce an oscillatory exit pressure at the exit of the test-section condenser and this is the cause, in Figs. 14a-14b, of somewhat reduced level of oscillatory behavior of other test-section flow variables. As a result of the instability in the auxiliary condenser, the dotted curve-A in Fig. 10 is merely suggestive of the possible presence of system instabilities. This is because the actual onset of oscillatory conditions has only a very indirect and incomplete relation to test-section M in and ∆T values used in Fig. 10. The issues regarding start-up time for the flow loop are discussed in Narain et al. [34] and Kurita [33]. 30 Oscillatory Category II Partial Condensation Flow . . . Exit Vapor Mass Rate MV Inlet Vapor Mass Rate Min Exit Liquid Flow Rate ML Test Section -P 1.7 0.5 in 1.5 0 1.3 -0.5 P 1.1 -1 0.9 -1.5 . MV . 0.7 -2 Test Section P (kPa) Mass Flow Rate (g/s) . M ML 0.5 -2.5 30 min. 0.3 -3 3 min. Time Fig. 14a: Time history depiction of M , M , M L and p values for an oscillatory partial unspecified exit condition case (category II). In Fig. 10, this flows’ appearance is indicated by crossing of the dotted curve-A. in V Oscillatory Category II Partial Condensation Flow Inlet Vapor Pressure (pin) Pressure(px6) Exit Pressure (pexit) Temperature at Rotameter(TR) 111 50 TR Pressure (Kpa) 107 49.5 49 pexit 105 48.5 103 48 101 Temperature (°C) 109 47.5 pin px6 99 47 30 min. 97 46.5 Time 3min. Fig. 14b: For the case in Fig. 14a, this figure shows the time history depiction of rotameter temperature TR and the following pressures: inlet pressure pin, pressure (px6) at location 6 in Fig. 7, and exit pressure pexit. 31 6. EXIT CONDITION PARAMETER AND SUMMARIZED GENERALIZATIONS OF RESULTS FOR THIS AND OTHER INTERNAL TWO-PHASE FLOWS As discussed above, pressure difference ∆p = pin – pexit in Fig. 13a, is typically not well defined for condensing duct flows just by the virtue of specified inlet and wall conditions alone - though it is well defined for single phase fluid flows. This is because outlet pressure often needs to be prescribed as a boundary condition to determine the flow field in the interior and the associated pattern of gas-liquid interfacial configurations in phase-change flows. The above stated conclusion for condensing flows, is also true, for many boiling flows - particularly if changes in interface location near the heat transfer surfaces (at specified wall temperatures) can be easily accomodated by significantly changing the interfacial mass-flux (and heat-flux) conditions. The reason behind the significance of exit pressure for these flows is that changes in exit pressure pexit tries to change the gas-phase squeezing pressure levels (pin + pexit)/2 and, concurrently, vapor motion due to different ∆p = pin - pexit values that are being imposed. Even small changes in these two factors (i.e. mean pressure level and pressure difference from inlet to exit of two-phase component) can be accomodated by phase-change flows because of the extra degree of freedom associated with various liquid-vapor interface configurations (flow morphologies) that are easily altered by changing the interfacial mass-flux (and associated wall heat-flux) assuming cooling/heating is done at fixed wall temperatures (not fixed heat-fluxes). However, historically, exit pressure specification has not been considered because of an incorrect and untested assumption that all two phase flow components behave similar to single phase flow components with regard to pressure drop across the component. As a result of this misunderstanding, for many “systems” in application, the pressure difference ∆p across a two-phase component gets inadvertantly specified or is subjected to large system imposed variations in ∆p. This leads to difficulties in experimentally characterizing two-phase flow behavior. Because of this, characterization of two-phase flow behavior of a condensing or flow boiling component in a system cannot easily be decoupled from the system and, therefore, due consideration should be given to the manner in which the component is integrated in the system hardware. All this requires a paradigm change from seeking a correlation for pressure difference ∆p to first assessing the value or range of externally imposed ∆p values. Only after this assessment, one can meaningfully develop and use suitable heat-transfer correlations. In the context of the results reported in section-1, when exit pressure is left unspecified (category II flow) through specially designed experimetnal arrangements, the flows further fall in the following two sub-categories, namely: (A) the ones for which a well defined steady “natural” pressure-difference “∆p|Natural” and associated “natural” interfacial configuration exist, and (B) the flows for which well defined steady “natural” pressure-difference “∆p|Natural” and associated “natural” interfacial configuration do not exist. An example of “natural” flow of type (A) is gravity dominated condensing flows or some upstream portions (x < x*) of shear driven flows. However, in practice, type (A) flows are seldom realized in gas-liquid flow components/test-sections unless the system is especially designed to offer freedom to choose exit pressures. This is because, if the condenser is not integrated in a specially designed system, an example of which is the system shown in Fig. 13b, then the flow is either unable to seek “∆p = ∆p|Natural” value due to exit pressure constraints or it gets inadvertently specified. Examples 32 of flow type (B) are fully condensing flows in micro-gravity or certain condensing flows in horizontal test sections (e.g. flows of the type in Fig. 3 with xe > x** or x* 0 because M in < M in − cr ). According to first principle based theory and supporting computational results (see [1]), these condensing flows, simply, cannot occur (over a range of inlet flow rates and duct lengths) in micro-gravity by self-selection of a steady exit pressure. p pf p= p p= p p f Natural Natural M in M in >M in-cr M=Min M Fig. 15a: This figure shows ‘ p|Natural vs M ’ curve for the condenser in Fig. 13a which is operating at a given value of temperature difference function T( ). The figure shows the operating vertical straight line AB that fixes the mass flow rate M = M in due to fixed values of heat load Q and fixed pressure pB for the boiler in Fig. 13a. The value p = p|Natural is achieved at intersection point D only for unspecified exit pressure condition when operating under the procedure O-1. Any other p = pf value can be fixed (see pf corresponding to point E above) with the help of operating procedure O-2. This is because for the specified exit condition cases (category I flows in [1]) pf value can lie anywhere in the shaded zone and is not required to lie on any particular curve. Operation of Specially Designed Experimental Arrangement for Condensing Flows Fig. 13a shows an special experimental arrangement designed and employed by us towards realizing both specified (category I) as well as unspecified (category II) exit conditions for condensing flow test section. The sytem and sub-system curves for the arrangement shown in Fig. 13a can be represented by the schematic shown in Fig. 15a. The undefined ∆p values for the condenser test section (at a specified value of vapor to condensing-surface temperature difference function ∆T( ), where is the coordinate that specifies the downstream locations of the condensing surface) in Fig. 13a are represented by the shaded zones in Fig. 15a. Within these shaded zones, there typically exists a zone (bounded by curves C1 and C2) within which the vapor flows have density variations within 1% of their inlet density values. 33 The mass flow rate M in = Q /hfg(pB) (where hfg is the heat of vaporization of the pure fluid employed in the system) is held fixed along line AB of Fig. 15a by the fixed heat load Q and a fixed pressure pB for the evaporator/boiler. In many practical applications, however, the line AB is a band which accounts for changing heat loads Q as well as changes in steady boiler pressure pB (which is often not held fixed by an accumulator). The unspecified pressure pexit case – procedure O-1 As stated earlier, pressure pA psat(TA) for nearly stagnant liquid in the accumulator A (in Fig. 13a) is allowed to vary in a range determined by the range of allowed temperatures TA. This degree of freedom for pressure pA is typically sufficient –except for some cases (see an example in [1]) - for the two-phase condensing flow to be able to vary its exit pressure “pexit” and seek its “natural” value “pexit|Natural.” As shown in Fig. 15a, this attainment of p|Natural is achieved at the intersection of the line AB and the well defined curve C that exists for type (A) flows in this category. The curve C represents a function p|Natural = f ( M in ) for a given inlet pressure pin and vapor to condensing-surface temperature difference ∆T( ). For each M in , the system design allows the flow in test section to have sufficient freedom in selection of p values and this is indicated by converging arrows at the intersection point D in Fig. 15a. Note that these curve C flows do not exist for many type (B) flows common to fully condensing microgravity situations (with xe > x** in Fig. 3) as no steady flow can be realized. The specified exit pressure pexit case – procedure O-2 Different pressure levels pA in the L-V separator of Fig. 13a are fixed by connecting it to the accumulator which employs active heating/cooling. This active heating/cooling of the accumulator is accomplished by a feedback control process using thermo-electric coolers and electric heaters controlled by suitable electronics, software, and a computer. These different fixed values of pA correspond to different fixed values of pexit and p p|Natural – and this is indicated by pf in Fig. 15a. Therefore resulting flows are in the specified exit pressure category (category I). This operational procedure allows us the choice to fix any value of pf leading to essentially incompressible vapor flow (if pf is fixed within the 1% compressibility band) or a flow with significant vapor compressibility (as shown in Fig. 15a). The compressible flows in this category require identification of another curve C4 in Fig. 15a – because flows to the top and right of C4 are likely to exhibit compressibility induced instabilities (such as oscillatory flows). As indicated earlier, for a 0g or horizontal category II flow in a condenser of a given length L, there exists M in-cr and hence a curve C3 in Fig. 15a needs to be demarcated. Only for flows to the top and right of this curve one has M in > M in-cr . Only under these conditions, one can ensure xe < x* for category II flows or suitably specify p for category I flows. Definition and Importance of Exit Pressure Parameter It is clear from the above discussion that for unspecified exit condition cases, type (A) condensing flows can be realized with pactual = p|Natural under a suitable experimental arrangement whereas type (B) flows will remain ill defined. On the other hand, for many condensing flows in the systems, p may be intentionally (as for Fig. 13a) or inadvertently (as for the system in Fig. 13b) specified. To 34 assess/predict condensing flows in this specified p category it is important to know the relation between pactual and a certain reference pressure p|Ref. The relationship between pactual and p|Ref. is characterized by a non-dimensional exit pressure parameter: pactual / p|Ref. (3) If the corresponding flow under unspecified exit condition is type (A) – that is p|Natural exists – we set p|Ref. = p|Natural. If the corresponding flow under unspecified exit condition is type (B) – that is p|Natural does not exist - we set p|Ref. = fknown ( Min ) to be a fixed number (e.g. 1 kPa) through a predefined function fknown of M in . It is recommended that the definition of this function fknown be chosen such that the resulting curve also lies (as does curve C) within the feasible curves (between curves C3 and C4) and furthermore, nearly incompressible zone bounded by the curves C1 and C2 in Fig. 15a. The above definition means that, for “natural” flows defined by type (A), the value of =1. Also, with the above choice of p|Ref., 1 signifies nearly incompressible flows for all specified exit pressure flow categories as well. pTotal p Pump p Total * p2 p2* p1* p1 p Natural p = p* p* M in M Fig. 15b: This figure shows the system and subsystem curves for condensing flow arrangement in Fig. 13b. The values pTotal*, p1*, p2* are decided by intersection of the line AB and various curves. p|Natural vs M curve for the condenser is the same curve shown in Fig. 15a that would have been obtained if the flow was realized under arrangement shown in Fig. 13a. The pressure drop across the condenser p* could be anywhere - its value being fixed by the system through the equation p* = pTotal* - ( p1*+ p2*). The band around p|Natural vs M curve shows that if p* falls within that band vapor compressibility effects will be small. 35 Determination of the Range of in Other Arrangements for Condensing Flows Fig 13b shows a representative arrangement for condensers employed in different applications where the exit condition is unknown and inadvertently specified. Instead of a displacement pump, a centrifugal pump is used to pump liquid flow back to the evaporator. The main difference between arrangements in Figs. 13a and 13b is that, in Fig. 13b, the accumulator-cum-L/V separator is missing in the liquid line between the condenser exit and the pump . This means as opposed to the arrangement in Fig. 13a, the two-phase flow in the condenser is not decoupled from single phase heavier fluid flow. The system and sub-system curves for the arrangement in Fig. 13b are shown by the schematic in Fig. 15b. Similar to the arragement in Fig. 13a, the mass flow rate Min in the system is assumed to be fixed along line AB shown in Fig. 15b (or in a narrow band around it) for fixed heat load Q and a fixed pressure pB for the boiler. As shown in Fig. 13b, the pressure difference between points B and P (as the flow goes through points I and E) is given by ∆pTotal System . This can be split in to pressure drops across sub-systems identified in Fig. 13b as: ∆pTotal System = p1 + p + p2 (4) where, as shown in Fig. 13b, p1 = pB –pin, p = pin –pexit , and p2= pexit - pP. Also, in Fig. 15b, for single phase fluid flows, p1 and p2 values are well defined functions of M because they are pressure differences for single–phase flow regions. For the two-phase condenser, pressure difference p is not a well defined function of M - and this fact is being represented by a shaded zone in Fig. 15b. Furthermore, in Fig. 15b, ∆pTotal Pump is the value of the pressure difference “pB -pP” as the flow goes across the pump. The ∆pTotal Pump curve in Fig. 15b is, however, obtained for the centrifugal pump by a separate characterization experiment in which the pressure difference “pB-pP” is measured as the liquid flows through the pump at different mass flow rates M . The steady mass flow rate M = Min achieved by this system is given by the intersection of line AB and the pump characterization curve ∆pTotal Pump in Fig. 15b. At this intersection point in Fig. 15b, ∆pTotal Pump = pTotal*. Furthermore, the system requires that at this Min , the pressure difference between points P and B be the same whether it is obtained by the flow (along B, I, E, and P) through the system or through the pump from P to B (as given by the pump curve). This means p = p* value actually achieved must be such that it satisfies (5) ∆pTotal Pump = ∆pTotal System leading to p*= pTotal* - p1* - p2* where p1* and p2* are also as shown in Fig. 15b. This means for the arrangement shown in Fig. 13b, the condition at the exit of the condenser is essentially a specified exit pressure flow condition (category I). In these kinds of practical arrangements, the value of is at the mercy of the system and is generally not known unless the above type of analysis is done and p*/ p|Ref is computed. If the condenser in Fig. 13b is properly sized to ensure that, over the range of values involved, the condenser actually achieves complete condensation then the remaining parts of the system in Fig. 13b can be made insensitive to the complexities of various flow regimes inside the condenser. For this special design, the curves represeting p1, p2 vs M in Fig. 15b must have high slopes and satisfy: | p1| ~ | p2| >> | p|. However, for proper designs of all such systems, one still needs to have good estimate of the range of values involved. Addition of an L/V separator (without the accumulator shown in Fig. 13a) at the exit of the 36 condenser section of Fig. 13b has also been used by others (see [37]). While this may allow attainment of natural unspecified exit condition steady flows, the start-up time is significantly larger (about 12 hours in [37] as opposed to about 20 minutes in [1]) than those associated with the system in Fig. 13a that employs a displacement pump. This is because the centrifugal pump in Fig. 13b seeks the correct mass flow rate Min while strongly affecting the pressure in L/V separator (without the accumulator). Impact of on Flow Regime Transition Boundaries for Condensing and Other Two-phase Flows It should be noted that the results described above are mainly applicable to condensing flows as these new ideas were developed by our group after extensive experimental/computational research on condensing flows. Even then, relevant ideas may also be extended to other two phase flows namely, boiling, adiabatic gas-liquid flows, etc. For a typical operating point P, in Fig. 15a and Fig. 15b, the entire condenser in the system is characterized by p, M , T( ) values. However, for flow within the condenser, for same set of these three parameters, there exist different flow regimes viz. annular, plug/slug, bubbly etc. Within the condenser, transition curves for these important flow regime boundaries (see [36]-[42]) – for given function T( ) - are generally defined as curves in “ M in - Z( )” space where vapor quality Z( ) is defined as a ratio of cross sectional vapor mass flow rate of M V to M in and takes a value of one at the inlet and zero at the exit. In the light of the results presented above, these transtion boundaries are significantly affected by the parameter and must be sought - for given function T( ) – as surfaces in “ M in - – Z( )” space. The above remarks about the impact of variable also clearly apply to flow boiling with sufficiently large exit vapor qualities. For adiabtic two-phase flows, though Z( ) is constant along in a two-phase test section, the flow regime boundaries are expected to vary because is expected to vary (in a small neighborhood of = 1) in “ M in - – Z( )” space. Though the variations in may be small from one experiment to another, the variations are expected to have a larger impact, particularly, on the location of the boundaries for the plug/slug and bubbly regimes. This may explain why ([42]) the available empirical correlations for p as a function of M in and Z( ) are good in the interior of flow regime boundaries but not so good on the more poorly defined flow regime boundaries. 7. CONCLUSIONS • • • This paper experimentally confirms the significance of exit conditions on the nature of quasi-steady internal condensing flows and proposes a novel and necessary exit-condition based categorization of these flows. In particular, for gravity driven condensate flows, experimental procedures and results for achieving steady and stable partial condensing flows under specified and unspecified exit conditions is presented here. In section 6, corresponding procedures for fully condensing flows are described. The experiments reinforce simulation results that, for partial condensation, multiple steady states, with quite different local and average heat transfer rates, are often achieved under different exit condition specifications (category I flows). Therefore, correlations for heat transfer coefficient (though not developed here) are only meaningful if flow regimes are clearly defined and developed in the framework of proposed exit-condition based categories. 37 • • • • • The existing simulation tool’s ability to be quantitatively correct in identifying “natural” exit conditions for gravity driven partial condensation cases under unspecified exit conditions (category II) is very good as this is supported by the reported experiments. This agreement adds credibility to the experimental results, simulation tool, and the proposed exit condition based categorizations. The transients in partial condensation experiments establish that the steady flows are definitely more robust under specified exit condition (category I) operation of condensers. This lends credibility to the simulation result that steady operation of shear driven condensers (in zero gravity and horizontal configurations) are much more difficult to achieve under unspecified exit conditions (category II). In general, these results suggest that specified exit-condition flows (category I flows) – for both partial and complete condensation cases - with properly selected exit pressure values are likely to be more robust and more readily realized than flows under other arrangements (category II or category III under arbitrarily fixed pressures and valve settings). Some flow regime and system boundaries for annular category I and category II flows are observed and reported here. Though these reported preliminary identifications of flow regime boundaries need to be made more definitive, their identifications are clearly important for attaining or ascertaining steady performances of condensers. This paper generalizes the above results and identifies a parameter , whose proper accounting is important for characterizing most forced internal duct phase-change flows. The experiments clearly demonstrate the difference between flow regime boundaries and systeminstability boundaries and the importance of identifying system-instability boundaries that are specific to individual systems. For example, curve A in Fig. 10 represents, system instability boundaries that arise from restrictions imposed on the exit pressure by phenomenon occurring in components downstream of the test section. 8. APPENDIX Channel Flow Governing Equations The liquid and vapor phases in the flow (e.g. see Fig. 1) are denoted by a subscript I: I = 1 for liquid and I = 2 for vapor. The fluid properties (density ρ, viscosity µ, specific heat Cp, and thermal conductivity k) with subscript I are assumed to take their representative constant values for each phase (I = 1 or 2). Let TI be the temperature fields, pI be the pressure fields, Ts (p) be the saturation temperature of the vapor as a function of local pressure p, ∆ be the film thickness, be the local interfacial mass flux, Tw ( ) (< Ts (p)) be a known temperature variation of the cooled bottom plate, and v = ˆi + ˆj I I I be the velocity fields. The flow fields are defined at every point x (a 3-D Euclidean position vector) and and be the components of gravity along x and y time t. Furthermore, let h be the channel height, axes, p0 be the inlet pressure, ∆T ≡ Ts (p0) - Tw (0) be a representative controlling temperature difference between the vapor and the bottom plate, hfg be the heat of vaporization at temperature Ts (p), and U be the average inlet vapor speed determined by the inlet mass flux. With t representing the actual time and ( , ) representing physical distances of a point with respect to the axes for the channel flow ( = 0 is at the inlet, = 0 is at the condensing surface, and = h is an isothermal slightly superheated noncondensing surface), we introduce a new list of fundamental non-dimensional variables – viz. (x,y, , ,u I ,v I , I , I ,m) - through the following definitions: 38 { , , ∆, I , } ≡ {h ⋅ x, h ⋅ y, h ⋅ , U ⋅ uI , 1 ⋅ U ⋅ m} { I , TI , pI , t} ≡ {U ⋅ vI , (∆ ) ⋅ I , p0 + I U2 ⋅ I , (h U) ⋅τ}. (A.1) Interior Equations The non-dimensional differential forms of mass, momentum (x and y components), and energy equations for incompressible flow in the interior of either of the phases are the well-known equations: ∂u I ∂v I + =0 ∂x ∂y ∂u I ∂u I ∂u I ∂π I 1 ∂ 2u I ∂ 2u I −1 + uI + vI =− + Frx + + ∂τ ∂x ∂y ∂x Re I ∂x 2 ∂y 2 1 ∂ 2 vI ∂ 2 vI ∂v I ∂v I ∂v I ∂ I −1 + uI + vI =− + Fry + + ∂τ ∂x ∂y ∂y Re I ∂x 2 ∂y 2 ∂ I ∂ ∂ 1 ∂2 I ∂2 I + u I I + vI I ≈ + ∂τ ∂x ∂y Re I PrI ∂x 2 ∂y 2 where Re I ≡ I Uh / −1 I , PrI ≡ I C pI / k I , Frx ≡ −1 x h / U 2 and Fry ≡ y (A.2) , h / U2. Interface Conditions The nearly exact interface conditions (see Delhaye [43]) for condensing flows are used. Utilizing a superscript “i” for values of flow variables at the interface (x,t) = − ∆ ( , t) = 0, the unit normal at any point on the interface, directed from the liquid towards the vapor, is denoted by n̂ and is equal to ∇ ∇ . Here, “ ∇ ” and “ ∇ S ” respectively denote 3-D (in Euclidean space) and surface 2-D (on curved surface of the interface) gradient/divergence operators depending on whether they are acting on scalar/vector fields. The unit tangent at any point on the interface, directed towards increasing , is denoted by t̂ . The velocity vs of a point on the dynamic interface is such that it satisfies the well known kinematic restriction: vs • n̂ = (−∂ / ∂t) / | ∇ | . Each phase is modeled as a viscous and incompressible T = −p I 1 + S I where viscous stress tensor Newtonian fluid with stress tensor T S I = I {(grad ⋅ v I ) + (grad ⋅ v I ) } / 2 and 1 is the identity tensor. The interface conditions and their nondimensional forms are given below. i.e. The tangential component of the vapor and liquid velocities at the interface must be continuous, v1i • tˆ = v i2 • tˆ . (A.3) The non-dimensional Cartesian form of the above requirement for the channel flow of interest becomes: 39 u i2 = u 1i − δ x ( v i2 − v1i ), (A.4) where δ x ≡ ∂δ / ∂x. Ignoring normal component of ∇ s σ and viscous stresses in the first equality, the second equality given below models the normal component of momentum balance at a point on the interface. That is: i i p1 = p 2 + 2 i (1/ 2 ≅ p2 + (1/ 2 − 1/ 1) + ∇ s • nˆ − ∇ sσ • nˆ + (S1i − S i2)nˆ • nˆ 2 − 1/ 1) − ( ∆ ) / [1+ ∆ 2 ] (A.5) 2/3 . The non-dimensional Cartesian form of the above requirement for the channel flow of interest becomes: i 1 = 2 i 2 − 1 1 We [1 + xx 2 3/ 2 x ] + m2 1 −1 , (A.6) 2 2 where We 1U h/σ, and surface tension σ is assumed to be nearly constant because of the nearly constant interface temperature. Because of smallness of surface tension force components and viscous stress components relative to the interfacial pressures, Eq. (A.6) with constant σ = σ (Ts (pi2 )) = σ (Ts (p )) is adequate. The tangential component of momentum balance at any point on the interface reduces to: Si = S1inˆ • tˆ = Si2nˆ • tˆ + ∇sσ• tˆ (A.7) The tangential component of momentum balance at the interface, as given by Eq. (A.7), becomes: ∂u1 ∂y i = 2 1 ∂u 2 i + [ term ] , ∂y (A.8) where the term [t] in Eq. (A.8) is defined as i [term] = { 2 1 i ∂ v 2 ∂ v1 2 − }+ [1 − ∂x ∂x i x i ∂u ∂v 2 { 1 − 1 }− 2 [1 − ∂y x ] ∂x i x i i ∂u ∂v ∂ 1 { 2 − 2 } + Ma 2 ⋅ 2 ∂y ∂x x ] 1 ∂x 1+ 2 2 . (A.9) x where the Marangoni number Ma ρ2Uc1d1/µ1 represents the surface tension contribution to tangential stress under the notation c1 ≡ dTs/dp and d1 ≡ -dσ/dT . For the cases considered here, a representative (for R113) set of constant values of c1 ≈ 0.0003 K/Pa, d1 ≈ 0.1046 N/(m-K), and Ma = 0.196 in Eq. (A.9) gives the same results as simulations without the Marangoni term. Hence the last Marangoni term on the right side of Eq. (A.9) can be dropped. The mass-fluxes V K and L K as determined by kinematic restrictions imposed by interfacial values of vapor and liquid velocities are: 40 VK The interfacial mass flux ≡− 2 Energy ( vi2 − vis ) • nˆ and LK (A.10) as obtained from energy balance at a point on the interface is given by: i E nergy ≡ − 1 ( v1i − vis ) • nˆ . i = 1/ h fg[{k1∇T1 • nˆ − k 2∇T 2 • nˆ } + 2 2 dσ 1 + { v1i − v is − v i2 − v is } dt s 2 + {S1inˆ • ( v1i − v si ) − S i2nˆ • ( v i2 − v is)}] i (A.11) i ∂ ∂ ≅ 1/ h fg[k1 T1 − k 2 T 2 ] . ∂n ∂n The non-dimensional form of kinematic mass fluxes i 1 VK and LK in Eq. (A.10) become: i 1 m LK ≡ u (∂ /∂x) − (v − ∂ /∂τ) / 1 + (∂ /∂x) 2 , and (A.12) m VK ≡ ( 2 / 1 ) u i2 (∂ /∂x) − (vi2 − ∂ /∂τ) / 1 + (∂ /∂x) 2 The non-dimensional form of Energy in Eq. (A.11) becomes: m Energy ≡ Ja /(Re1 Pr1 ){∂ 1 / ∂n i −(k 2 / k 1 )∂ where Ja ≡ Cp1 i 2 / ∂n } , (A.13) / h 0 fg , and h 0 fg ≡ h fg (Ts (p )) . The dimensional and non-dimensional forms of interfacial mass balance requirements respectively become: LK = VK = Energy ≡ . (A.14) m LK = m VK = m Energy ≡ m. (A.15) Under the assumption of negligible non-equilibrium thermodynamic effects and, therefore, negligible interfacial resistance (a valid assumption for all away from ~ 0 because interfacial massflux rapidly drops to modest values), the interfacial temperature values satisfy: i i i (A.16) T1 ≅ T 2 = T s(p 2) . The non-dimensional thermodynamic restriction on interfacial temperatures, as given by Eq. (A.16), becomes: i 1 ≅ i 2 = S (p ) i 2 ≡ s ( ). i 2 (A.17) Within the vapor phase, for the refrigerants considered here, changes in absolute pressure relative to the inlet pressure are typically so small that s ( i2 ) ≅ s (0) constant is a good approximation. Boundary Conditions The problem posed by Eqs. (A.2), (A.4), (A.6), (A.8), (A.15), and (A.17) are computationally solved subject to boundary conditions that are 41 At the inlet (x = 0, 0 y 1) at any time τ: u 2 (0,y,τ) = 1 2 (0,y,τ) = 0 At the bottom wall (y = 0, 0 x v 2 (0,y,τ) = 0 2 (0,y,τ) = s (A.18) (0) . xe) at any time τ: u1 (x, 0, τ) = 0 , v1 (x, 0, τ) = 0 , 1 (x, 0, τ) = , w (A.19) where w ≡ w ( ) is a constant unless it is otherwise specified. In case of flow in Fig. 2, this situation arises whenever, for a given heat load, the coolant flow rate is high enough to make the coolant temperature rise negligible as it flows past the test-section. At the top wall (y = 1, 0 x xe) at any time τ: u 2 (x,1, τ) = 0 , v 2 (x,1, τ) = 0 , 2 (x,1, τ) = s (0) . (A.20) Furthermore, because of the nature of boundary conditions in Eqs. (A.18) – (A.20), 2 (x, y, τ) ≅ s (0) is assumed/prescribed to limit the discussions in this paper to the flow of saturated vapor. This is done because, for the pure vapor flows considered here, it is easy to verify the well-known fact that the effects of superheat (commonly in the 5 – 10 oC range) are negligible. Exit Conditions Any condenser section of the type shown in Fig. 1 is typically a part of a closed flow loop. A flow loop which maintains a steady condensing surface temperature Tw( ), a constant flow rate, and a constant pressure p (i.e. π2 = 0) at the inlet, may also be designed to provide: (a) specified steady exit pressure (category-I flows) which is equivalent to, for partial condensation flows, a specified value of steady exit quality Ze (the ratio of vapor mass flow rate at exit, x = xe, to vapor mass flow rate at inlet), and (b) unspecified exit pressure (category-II flows) which is equivalent to, for partial condensation flows, unspecified values of exit quality Ze (t). Category-I flows: For this case of specified exit conditions, it is assumed that the exit pressure or the exit quality Ze specifications are still such that vapor compressibility effects can be ignored. The boundary value problem (BVP) constituting of the steady governing equations and boundary conditions – including specified steady exit conditions (steady Ze or pexit) – constitutes a steady “well-posed” category I flow problem. Steady category-I flow solutions of the governing equations, obtained by dropping all time dependencies in all the relevant equations, and solving the resulting steady equations (which are elliptic) for steady prescribed values of Ze are presented here. The prescription of Ze within 0 < Ze< 1 is arbitrary except that it should be such that a steady computational solution in the assumed stratified/annular regime be feasible. The initial boundary value problem (IBVP) for the unsteady governing equations with arbitrary initial guesses under specified steady exit conditions are also “well-posed.” These solutions are supposed to be such that they get attracted, as , to the corresponding steady solution. If not, the unsteady solutions’ behavior at large τ indicates the stability of the corresponding steady solution. Such stability analyses are reported, for h external flow problems of Nusselt [21] and Koh [29], in Phan 42 and Narain ([28]) and Kulkarni et al. ([2]-[3]). Such stability analyses for steady internal category I flows, however, are not currently available and are the subjects of forthcoming papers. An inspection of all the non-dimensional governing equations, interface conditions, and boundary conditions reveal the fact that, for category I flows, the flow is affected by the following set of non-dimensional parameters: {Rein , Ja, Fr −x1 , 2 , 2 1 , Pr1 , x e , Z e , We, Fr −y1}, (A.21) 1 where Rein ρ2Uh/µ2 Re2. Here Rein, Frx-1, and Ja are control parameters associated with inlet speed U, channel inclination α that determines the components of gravitational acceleration, and temperature difference T. Clearly the value of Ze or exit pressure is important for this problem. The density ratio ρ2/ρ1, viscosity ratio µ2/µ1, and Prandtl number Pr1 are passive fluid parameters. Also, for unsteady or quasi-steady wavy-interface situations, the above equations imply additional dependences on a surface -1 2 2 tension parameter, Weber number We 1U h/σ, and a transverse gravity parameter Fry yh/U . For superheated vapors, there is a very weak dependence, through Eq. (A.13), on the thermal conductivity ratio k2/k1. Category-II flows: For these flows, no exit conditions are prescribed. The steady governing equations and the remaining boundary conditions constitute an “ill-posed” boundary value problem (BVP) and no unique solution is possible. However, when unsteady category-II flows are considered for suitable initial condition choices, the initial boundary value problems (IBVP) are ‘well-posed.” However, in general, as , no well-defined steady solution may exist. If they do, as is the case for gravity driven flows, one obtains a unique well defined “attracting” solution that is independent of initial guesses and one also obtains a corresponding unique exit condition Ze (τ) . Ze|Na as τ . . The stability of these attracting flows to initial disturbances has been extensively considered in [4] – [6]. For shear driven internal flows in this category, the existence or non-existence of “attractors” follow the pattern discussed for Fig. 3. Note τ = 0 cannot be chosen to be the time when saturated vapor first comes in contact and condenses on a dry sub-cooled (Tw ( ) < Ts (p0)) bottom plate. This is because the above described continuum equations will not apply at early times (τ ~ 0) because inter-molecular forces will be important for very thin (approximately over 10 - 100 nm of film thickness) condensate film δ(x, τ) . Because of the above modeling limitations, the strategy here is to start at τ = 0, with any sufficiently thick guess and look at the long time τ behavior of the solution. Most of the time, for τ = 0 initial guess, we choose steady solution of a category-I flow. That is, if φ(x,y,τ) is any variable (such as uI, vI, πI, θI, etc.), the initial values of φ and film thickness δ(x,τ) are such that: φ( x , y,0) = φsteady ( x , y) and δ( x ,0) = δsteady ( x ) , (A.22) where φsteady and δ steady are solutions of the category-I problem for a certain exit condition (exit quality Ze or exit pressure). 43 9. ACKNOWLEDGMENT This work was supported by the NSF grant CTS-0086988 and NASA grant NNC04GB52G. 10. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Narain, A., J. H. Kurita, M. Kivisalu, A. Siemionko, S. Kulkarni, T. Ng, N. Kim, and L. Phan, Oct. 2007, “Internal Condensing Flows inside a Vertical Pipe – Experimental/Computational Investigations of the Effects of Specified and Unspecified (Free) Conditions at Exit,” Journal of Heat Transfer, Vol. 129, pp. 1352-1372. S. D. Kulkarni, A. Narain, S. Mitra, and L. Phan, 2008, “Forced Flow of Vapor Condensing over a Horizontal Plate (Problem of Cess and Koh*) Part I – Steady Solutions of the Full 2D Governing Equations,” Being submitted by Oct. 10, ‘07 (after good IMECE07 reviews) for publication in ASME Journal of Heat Transfer. S. D. Kulkarni, A. Narain, and S. Mitra, 2008, “Forced Flow of Vapor Condensing over a Horizontal Plate (Problem of Cess and Koh*) Part II – Unsteady Solutions of the Full 2D Governing Equations Yielding Steady “Attractors” and their Unsteady Responses to Initial Disturbances and Noise,” Being submitted by Oct. 10, ‘07 (after good IMECE07 reviews) for publication in ASME Journal of Heat Transfer. Narain, A., Q. Liang, G. Yu, and X. Wang, Jan. 2004, “Direct Computational Simulations for Internal Condensing Flows and Results on Attainability/Stability of Steady Solutions, their Intrinsic Waviness, and Their Noise-sensitivity,” Journal of Applied Mechanics, Vol. 71, pp. 69-88. Liang, Q., X. Wang, and A. Narain, October 2004, “Effect of Gravity, Shear and Surface Tension in Internal Condensing Flows - Results from Direct Computational Simulations.” ASME Journal of Heat Transfer, 126 (5), pp. 676-686. Phan, L., X. Wang, and A. Narain, 2006, “Exit condition, Gravity and Surface-Tension Effects on Stability and Noise Sensitivity Issues for Steady Condensing Flows inside Tubes and Channels,” International Journal of Heat and Mass Transfer, Vol. 49, Issues 13-14, pp.2058-2076. Goodykoontz, J. H. and R. G. Dorsch, 1966, “Local heat transfer coefficients for condensation of steam in vertical down flow within a 5/8-inch-diameter tube,” NASA TN D-3326. Goodykoontz, J. H. and R. G. Dorsch, 1967, “Local heat transfer coefficients and static pressures for condensation of high-velocity steam within a tube,” NASA TN D-3953. Carpenter, F. G., 1948, “Heat transfer and pressure drop for condensing pure vapors inside vertical tubes at high vapor velocities,” Ph. D. thesis, University of Delaware. Yu, G., 1999, “Development of a CFD Code for Computational Simulations and Flow Physics of Annular/Stratified Film Condensation Flows,” Ph.D. Thesis, ME-EM Department, Michigan Technological University. Rabas, T. J., and B. Arman, 2000, “Effects of the Exit Condition on the Perofromance of In-Tube Condensers,” Heat Tranfer Engineering, 21(1), pp. 4-14. Wedekind, G. L. and B. L. Bhatt, 1977, “An Experimental and Theorotical Investigation in to Thermally Governed Transient Flow Surges in Two-Phase Condensing Flow,” ASME Journal of Heat Transfer 99 (4), pp. 561 – 567. Bhatt, B. L. and G. L. Wedekind, 1980, “Transient and Frequency Characteristics of Two-Phase Condensing Flows: With and Without Compressibilty,” ASME Journal of Heat Transfer 102 (3), 495-500. Bhatt, B. L. and G. L.Wedekind, 1980b, “ A Self Sustained Oscillatory Flow Phenomenon in Two- 44 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. Phase Condensing Flow Systems,” ASME Journal of Heat Transfer 102 (4), 695-700. Wedekind, G. L. and B. L. Bhatt, 1989, “ Modeling the Thermally Governed transient Flow Surges in Multitube Condensing Flow Systems with Thermal Flow Distribution Asymmetry,” ASME Journal of Heat Transfer 111 (3), pp. 786 –791. Bhatt, B. L., G. L. Wedekind, and K. Jung, 1989, “ Effects of Two-phase pressure Drop on the SelfSustained Oscillatory Instabilty in Condensing Flow,” ASME Journal of Heat Transfer 111, 538545. Boyer, D. B., G. E. Robinson, and T. G. Hughes, 1995, “Experimental Investigation of Flow Regimes and Oscillatory Phenomena of Condensing Steam in a Single Vertical Annular Passage,” International Journal of Multiphase Flow, Vol. 21, No. 1, pp. 61 – 74. Wedekind, G. L., C. J. Kobus, and B. L. Bhatt, 1997, “Modeling the Characteristics of Thermally Governed Transient Flow Surges in Multitube Two-Phase Condensing Flow Systems with Compressibility and Thermal and Flow Distribution Asymmetry,” ASME Journal of Heat Transfer, Vol. 119, No. 3, pp. 534 – 543. Kobus, C. J., G. L. Wedekind, and B. L. Bhatt, 2000, “Predicting the Influence of Compressibility and Thermal and Flow Distribution Asymmetry on the Frequency-Response Characteristics of Multitube Two-Phase Condensing Flow Systems,” ASME Journal of Heat Transfer, Vol. 122, No. 1, pp. 196 – 200. Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington D. C. Journal of Multiphase Flow, Vol. 16, pp. 139 – 151. Nusselt, W., 1916, “Die Oberflächenkondesation des Wasserdampfes,” Z. Ver. Dt. Ing. 60 (27), pp. 541-546. Narain, A. and Y. Kizilyalli, 1991, “Pressure driven flow of pure vapor undergoing laminar film condensation between parallel plates,” International Journal of Non-Linear Mechanics, Vol. 26, No. 5, pp. 501-520. Narain, A., G. Yu, and Q. Liu, 1997, “Interfacial Shear Models and Their Required Asymptotic Form for Annular Film Condensation Flows in Inclined Channels and Vertical Pipes,” International Journal of Heat and Mass Transfer, Vol. 40, N0. 15, pp. 3559 - 3575. Garimella, S., J. D. Killion, and J. W. Coleman, 2002 “An Experimentally Validated Model for Two-Phase Pressure Drop in the Intermittent Flow Regime for Circular Microchannels,” Journal. of Fluids Engineering, Vol. 124, pp. 205-214. Garimella, S., J. D. Killion, and J. W. Coleman, 2003 “An Experimentally Validated Model for Two-Phase Pressure Drop in the Intermittent Flow Regime for Non-Circular Microchannels,” Journal of Fluids Engineering., Vol. 125, pp. 887-894 Liao, N. S., C. C. Wang, and C. L. Tien, 1988, "Analysis of Transient Flow Surge Phenomena in a Single-Tube Condenser,” International Communications in Heat and Mass Transfer, Vol. 15, pp. 257 – 268. Liao, N. S. and C. C. Wang, 1990, “Transient response Characteristics of Two-Phase Condensing Flows,” International Journal of Multiphase Flow, Vol. 16, 139-151. Phan, L. and A. Narain, 2007, “Non-linear Stability of the Classical Nusselt problem of Film Condensation and Wave Effects,” ASME Journal of Applied Mechanics, Vol.74, No.2, pp. 279-290. Koh, J. C. Y., 1962, “Film Condensation in a Forced-Convection Boundary-Layer Flow,” International Journal of Heat and Mass Transfer, Vol. 5, pp. 941-954. Palen, J. W., R. S. Kistler, and Y. Z. Frank, 1993, “What We Still Don’t Know About Condensation in Tubes,” In Condensation and Condenser Design (Edited by J. Taborek, J. Rose and I. Tanasawa), Pub.: United Engineering Trustees, Inc. for Engineering Foundation and ASME, New York, pp. 19– 45 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 53. Ng, T. W., 2006, “Development and Calibration of a Fluorescence and Fiber-Optics Based RealTime Thickness Sensor for Dynamic Liquid Films,” Ph.D. Thesis, MEEM, Michigan Technological University. Siemonko, A., 2006, “Design, Fabrication, and Operation of a System to Control FC-72 Condensation Inside a Vertical Tube,” Ph.D. Thesis, Chem Eng, Michigan Technological University. Kurita, J. H., 2006, “Experimental Investigation of Fully Condensing Downward Vapor Flows in a Vertical Tube - Unspecified (Free) Exit Condition Cases,” M. S. Thesis, MEEM, Michigan Technological University. Narain, A., J. H. Kurita, M. Kivisalu, A. Siemionko, S. Kulkarni. T. Ng., N. Kim, and L. Phan, Jan 2007, “Internal Condensing Flows inside a Vertical Pipe – Experimental/Computational Investigations of the Effects of Specified and Unspecified (Free) Exit Conditions at Exit,” A Technical Report which is an extended version of ref. [1] and is available at: http://www.me.mtu.edu/~narain/narainpublications1.htm Parratt, L. G., 1961, Probability and Experimental Errors in Science – An Elementary Survey, Wiley, New York. Carey, V. P., 1992, Liquid-Vapor Phase-Change Phenomena, Series in Chemical and Mechanical Engineering, Hemisphere Publishing Corporation, New York. Lu, Q. and N. V. Suryanarayana, 1995, “Condensation of a Vapor Flowing Inside a Horizontal Rectangular Duct,” Journal of Heat Transfer, 117, pp. 418-424. Hewitt, G. F., Shires, G. L., and Bott, T. R., 1994, Process Heat Transfer, CRC Press, Inc., Begall House, Boca Raton, Florida. Lockhart, R. W. and Martinelli, R.C., 1949, “Proposed Correlation of Data for Isothermal TwoPhase two-component flow in a pipe,” Chem. Eng. Prog., 45, No. 1, pp. 39–48. Taitel, Y. and Dukler, A. E., 1976, “A Model for predicting Flow regime transition in horizontal and near horizontal gas-liquid flow,” AIChE Journal, 22, pp. 47-55. Chen, J. C., 1966, “Correlation for boiling heat transfer to saturated fluids in convective flow,” Ind. Chem. Eng. Proc. Design and Dev., 5, pp. 322-339. McDermott, B. and Timothy Shedd, “Detailed Pressure Drop and Flow Regime Characteristics of Adiabatic Two-Phase Flow of Pure Refrigerant R-123,” Paper no. IMECE2007-43829, Proceedings (CD ROM) of ASME, IMECE 2007, Seattle, Nov. 11-15, 2007. Delhaye, J. M., 1974, “Jump Conditions and Entropy Sources in Two-phase Systems; Local Instant Formulation,” Int. J. of Multiphase Flow, 1, pp. 395-409. 46