Effect of wall surface roughness on condensation shock

International Journal of Thermal Sciences 132 (2018) 435–445

Contents lists available at ScienceDirect

International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Effect of wall surface roughness on condensation shock ∗

T

Aditya Pillai , B.V.S.S.S. Prasad Turbo Machines Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

A R T I C LE I N FO

A B S T R A C T

Keywords: Condensation shock Surface roughness Boundary layer Non-equilibrium Wet steam Nozzle Two-phase flow

The effect of sand grain type surface roughness on an unseparated incipient condensation shock and its influence on the boundary layer has been studied numerically for wet steam supersonic flow in a Laval nozzle. In the vicinity of the condensation shock, the sudden jump in density and a reduction in the wall shear stress causes a rapid drop in skin friction. The pressure shows a mild adverse gradient in this region. The boundary-layer velocity profiles vary with roughness value at a constant inlet pressure. The boundary layer thickness increases with the increase in either the inlet pressure, or the size of the roughness elements. As the roughness height is varied from 1 μm to 1000 μm the boundary layer thickness increases typically by 33% whilst the peak value of pressure gradient drops down by 50%. The boundary layer thickness is found to be maximum for po = 80 kPa owing to secondary nucleation. The peak value of pressure gradient is noticed to be doubled as the flow medium changes from dry air to wet steam.

1. Introduction In the last few stages of steam turbines with large power output, the presence of wet steam influences the condition line crossing over into the two-phase region. As this happens, condensation takes place, not as soon as it crosses the saturation line, but only after it has attained some degree of supercooling. The point at which the liquid droplets are formed is called the Wilson point. The steam is in metastable state and the condensation occurs in non-equilibrium. Latent heat is released from liquid water as the phase change occurs and this heat has to be taken over by the surrounding vapour phase. This raises the temperature of the vapour which in turn increases its pressure. This sudden increase in pressure formed due to condensation, is called condensation shock. Fig. 1 shows the pressure variation along the nozzle of Moses and Stein [1] (a) when the working medium is dry air and (b) when it is wet steam. The two curves in Fig. 1 portray the relative effect of condensation shock on the flow phenomenon. After the condensation shock the steam generally reverts to near thermodynamic equilibrium where the temperatures of both the vapour and the droplets are close to the saturation level. Since the growth of liquid phase takes place by heat transfer through finite temperature difference between the phases, the process is essentially irreversible. The net rise in entropy that appears as a reduction in the potential for performing work can be equated to thermodynamic wetness loss, a reason for loss in turbine efficiency. If the degree of subcooling is increased further, an aerodynamic shock wave may be formed inside the condensation zone. Inlet ∗

subcooling in the present study does not exceed this limiting value and the pressure rise observed here is gradual, so it can be said that the flow is independent of any aerodynamic shock and only condensation shock can be seen. The presence of condensation leads to problems of blade erosion and losses in turbine efficiency. Wet steam two-phase condensing flows usually occur in the supersonic flow conditions. Condensation shock is a major feature of the non-equilibrium condensation phenomenon. Condensation shock represents nucleation of droplets in the flow and their subsequent growth. Of the three main components of wetness loss, thermodynamic losses are associated with the irreversible heat transfer between the two phases during condensation. Numerical studies are performed with an attempt to capture condensation shock and to estimate the thermodynamic losses [2]. Emphasis on condensation shock to alleviate the wetness losses have been made. Parametric studies have been conducted to determine what influences condensation shock and how these strategies can be implemented to reduce the thermodynamic losses. At first, the necessary conditions for the existence of condensation shocks were studied [3] and further efforts were made to control and suppress condensation shocks [4–6]. Shock strength reduction indicates a reduction in entropy and hence thermodynamic losses. It is important to study the influence of different parameters on non-equilibrium condensation so that losses can be estimated and mitigated more accurately. However, the effect of surface roughness on non-equilibrium condensation has not been studied yet. During the operation of steam turbines, the blade surfaces experience severe performance degradation. Erosion of blades due to heat,

Corresponding author. E-mail address: [email protected] (A. Pillai).

https://doi.org/10.1016/j.ijthermalsci.2018.06.028 Received 26 October 2017; Received in revised form 28 January 2018; Accepted 21 June 2018

1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

Nomenclature Cf Cμ CS C.V. co Dω E Gk, Gω H hlv I Ks+ Ks k L Mm P˙ p po pout qc r r* S Sk, Sω To Td TG TR

up u* u+ Vd x Δx Yk, Yω yp y+ y Δy

Skin friction coefficient Constant in modelling turbulent viscosity Roughness constant Control Volume Speed of sound Cross diffusion term Total Energy per unit mass Generation term for k and ω Total Enthalpy Specific enthalpy of evaporation at pressure p Nucleation rate Non-dimensional roughness height Equivalent sand grain roughness Turbulence kinetic energy Nozzle Length Mass of 1 molecule Expansion rate Local Pressure Inlet Total Pressure Static pressure at exit Evaporation coefficient Average radius of droplet Critical droplet radius Super saturation ratio User defined source terms for k and ω Inlet stagnation temperature Droplet temperature Vapour temperature Reduced temperature

Velocity parallel to the wall Friction velocity Non-dimensional height, up/u* Average droplet volume Axial distance Minimum cell width Dissipation term for k and ω due to turbulence Distance from the wall Non-dimensional height, (u*.y/ν) Height Minimum cell height

Greek symbols β δ η Γ Γk, Γω ϒ κB μ μt ν ω ρ, ρl, ρv σ θ τw

Mass fraction of condensed liquid phase Boundary layer thickness Number density of droplets per unit volume Mass generation rate due to condensation and evaporation Effective diffusivity term for k and ω Ratio of specific heat capacities Boltzmann constant Dynamic viscosity Turbulent or Eddy viscosity Kinetic Viscosity Specific dissipation rate Density of mixture, liquid & vapour phase evaluated at temperature T Liquid surface tension Non-isothermal correction factor Wall shear stress

predict the wet steam condensation phenomenon [7–16]. Homogeneous spontaneous condensation of steam is still being studied with an intention of increasing the last stage and overall efficiency of the turbine [17]. Since the devices in which the condensation process takes place are extremely complex, it makes it quite difficult to model and conduct numerical and experimental research. On the other hand, a simple model for simulating complex flows in practical domains is a converging-diverging nozzle. In the past, different nozzles each with different characteristics, were designed and experiments were performed to study the wet steam condensation (ex. Moore [18], Moses [1], Binnie [19], Gyarmathy [20]). Different parameters that perturb the flow physics and influence the losses were investigated. Early works [21,22] include the study of the structure and growth of the turbulent boundary layers once it encounters roughness and discuss the conditions that should be met for self-preservation. Liu and Squire [23] studied shock wave boundary layer interactions (SWBLIs) on curved surfaces and concluded that the critical peak Mach number does not change very much with the surface curvature and is close to 1.30. Babinsky and coworkers [24–26] conducted extensive investigations to determine the effects of surface roughness on turbulent boundary layers. The velocity profiles downstream of the roughness were found to be less full, and skin friction was reduced but no large-scale separation due to the additional effects of roughness was observed. They also found that, even for roughness heights in the hydraulically smooth regime, incoming boundary layers were thicker, and slightly less full than smooth wall profiles. Theoretical considerations also suggested an influence of surface roughness on incipient separation. The effect of irregular surface roughness on turbulent boundary layer was considered by Wu and Christensen [27] which indicate that large-scale low and high-momentum regions exist in both smooth and rough wall flows and these large-scale features contain a lot of energy and manifest a majority of the Reynolds shear stress.

(i.e., thermal erosion) the collision of particles or impurities and deposition of impurities on the blades surface significantly affect the surface roughness of the blades. A large quantity of experimental and numerical work has been done on the performance losses of turbines due to blade surface roughness. Early investigations into the performance of wet steam turbine stages were endeavors to better understand the losses associated with non-equilibrium condensation. Experiments were performed on turbine rotor cascades and modelling approaches were examined to better

Fig. 1. Pressure variation along the nozzle length with and without shock at po = 90 kPa. 436

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

interaction with the boundary layer on a curved surface has been provided in the present work. The condensation shock/boundary-layer interactions with boundary layers on a wall with a curvature are still not well understood. This paper presents the results of a detailed numerical investigation of such flows. The present study focuses on the condensation shock interactions with the boundary layer parameters such as boundary layer thickness (δ). The flow variables such as pressure, pressure gradient, skin friction coefficient (Cf) and the boundary layer velocity profiles are presented. The effects of surface roughness on condensation shock and its interaction with the boundary layer is reported. Inlet subcooling is limited to obtain condensation shock and presence of aerodynamic shock is not considered.

Squire [28] and Townsend [29] reported modelling by sand grain roughness to observe the turbulent boundary layers for high Reynolds number. Strong logarithmic dependence of roughness values on stream wise velocity was found and the slopes of these logarithmic profiles were same for both smooth and rough wall cases illustrating similar dynamics in the region. Their results show a collapse of stream wise mean velocity defect and skewness profiles in the outer region of the flow and that the velocity variance indicates a dependence on equivalent sand grain roughness Reynolds numbers. It is almost impossible to avoid rough surfaces in turbomachinery flows and boundary layers encounter roughness very often. Taylor [30] and Bons [31] considered the surface roughness measurements on gas turbine blades to gain insights into statistical characteristics of the surface roughness and found that all regions showed roughness levels 4 to 8 times greater than the levels for production line hardware. Flow history is important when considering roughness effects on boundary layer as roughness experienced upstream of the flow may also affect the physics [32]. It is also understood that rough walls prevent flow reversion and a faster increase in the turbulence intensity near the wall [33]. Step changes in roughness upstream of the boundary layer shows that above the internal layer the flow exhibits characteristics of a rough, wall-bounded flow, whereas near the wall the turbulence intensity is similar to that of an isolated smooth wall [34]. Roughness can even trigger transition in the boundary layer and may generate bow shock [35] and the transition also depends on the shape of these roughness elements [36]. Some recent studies [37] have made an effort to better understand the condensation of steam in boundary layers. The inspections suggest that viscous dissipation and reduced expansion rate within a typical boundary layer influence nucleation and growth, leading to droplet radii and size distributions that differ substantially from those predicted in inviscid flow. Wet steam flow models are validated using different condensing nozzles and most studies performed do not assume the role of boundary layers in affecting the steam flow and droplet generation, whereas Starzmann [38] has shown that the pressure distribution is significantly affected by the additional blockage due to wall boundary layer. He adds that boundary layer effects impact the mean nozzle flow and thus influences the validation process of condensation models. Nicol and Medwell [39] stated that surface roughness influences the condensation of steam, its heat transfer coefficient and droplet deposition mechanics. The calculations considered in the present study take into account the wall curvature [40] and there is potential for a significant addition to the literature on the subject. Schnerr [41] and Setoguchi [42] were amongst the first people to study the effect of nonequilibrium condensation on aerodynamic SWBLIs. However, the effects of condensation shock in Laval nozzles with surface roughness are not investigated to date. Thus, there have been many studies concerning roughness effects on boundary layer flows. Non-equilibrium condensation has also been investigated by many researchers over the decades and attempts to answers the questions like what influences non-equilibrium condensation, how condensation shocks are controlled and how they change the flow behaviour, what are the losses associated with it and how the performance of turbines are influenced by these parameters were made. However, questions like how the flow field changes when condensation shock interacts with the boundary layer, how surface roughness affects these interactions and how these interactions take place on the curved surface of a nozzle are yet to be answered. The present work aims to answer these questions and presents the results of a detailed numerical investigation of such flows. The quantitative information about the effect of surface roughness on condensation shock and its interaction with the boundary layer on the curved surface of a nozzle has been provided. Thus, the shock/boundary-layer interactions with boundary layers developing under non-zero pressure gradient, on a wall with a curvature are still not well understood. The quantitative information about the effect of surface roughness on condensation shock and its

2. Governing equations Condensation is a phase transition phenomenon that occurs often when a radical expansion takes place. The wet steam model implemented by Ansys Fluent flow uses two additional transport equations in conjunction with the compressible Navier-Stokes equation. The additional equations govern the mass fraction of the condensed liquid phase and models the evolution of the number density of the droplets per unit volume. The set of equations governing the wet steam flow are presented below. The flow of the mixture is governed by the compressible form of the Navier-Stokes equations given in vector form by

∂W ∂ ∂Q ∂t

∫ QdV∮ [F − G]. dA = ∫ HdV . V

V

(1)

Q is the heat flux. Total energy E is related to total enthalpy H by the relation

E=H−

p ρ

(2)

The transport equation governing the mass fraction of the condensed liquid phase, β, is given by

∂ρβ v β) = Γ + ∇ . (ρ→ ∂t

(3)

The transport equation that models the evolution of the number density of the droplets per unit volume is given by

∂ρη v η) = ρI + ∇ . (ρ→ ∂t

(4)

To determine the number of droplets per unit volume, mixture density and the average droplet volume are combined in the following expression

β (1 − β ) Vd (ρl / ρv )

η=

(5)

where ρl is the liquid density and the average droplet volume is defined as

Vd =

4 3 πrd 3

(6)

The mass generation rate Γ in the classical nucleation theory during the non-equilibrium condensation process is given by the sum of mass increase due to nucleation and also due to growth/demise of these droplets by the equation

Γ=

4 ∂r π ρ Ir*3 + 4π ρl ηr 2 . 3 l ∂t

(7)

Where r is the average radius of the droplet, and r* is the KelvinHelmholtz critical droplet radius, above which the droplet will grow and below which the droplet will evaporate. An expression for r* is given by 437

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

2σ ρl RTlnS

r* =

∂ ∂ ∂ ⎛ ∂ω ⎞ (ρω) + (ρωuj ) = ⎜Γω ⎟ + Gω − Yω + Dω + Sω ∂t ∂x j ∂x j ⎝ ∂x j ⎠

(8)

where σ is the liquid surface tension evaluated at temperature T, ρl is the condensed liquid density evaluated at temperature T and S is the super saturation ratio defined as the ratio of vapour pressure to the equilibrium saturation pressure as

P Psat (T )

S=

Γk and Γω represent the effective diffusivity of k and ω respectively. Gk and Gω represent the production of turbulence kinetic energy and the generation of ω. Yk and Yω represent the dissipation of k and ω due to turbulence and Dω represents the cross diffusion term. Sk and Sω are user defined source terms. The model constants required to compute the * above terms were taken to be α∞ = 1, α∞ = 0.52, β∞* = 0.09 and a1 = 0.31. The method chosen to specify turbulent quantities were to provide the turbulent intensity and viscosity ratio at the inlet and outlet. The inlet and backflow turbulent intensity was set to 5% whereas the inlet and backflow turbulent viscosity ratio was set to 10. No limiting conditions for turbulent quantities on the wall were specified. For effectively modelling the surface roughness effects in a turbulent wall bounded flow, the law-of-the-wall modified for roughness is used. Experiments in roughened pipes and channels indicate that the mean velocity distribution near rough walls, although having the same slope (1/κ), shift from their regular position giving a different intercept for different surface roughness values (additive constant B in the loglaw). Therefore, the law-of-the-wall for mean velocity modified for roughness has the form

(9)

The surface tension ( σ ) of water is a function of temperature only and an equation accurate to within 0.3% for temperature below 280 degrees Celsius is used to model the surface tension in the present study and is given by

σ = 82.27 + 75.612TR − 256.889TR2 + 95.928TR3

(10)

where TR is the reduced temperature given by

TR =

TG 647.29

(11)

and TG is the vapour temperature. The mechanism of the transfer of mass from the vapour to the droplets and the transfer of heat from the droplets to the vapour in the form of latent heat is given by an energy transfer relation written as

u* =

(13)

(14)

It is assumed that the mixture density ρ is related to the vapour density ρv by the following equation

ρ ρ = ⎜⎛ v ⎞⎟ 1 β⎠ − ⎝

is the friction velocity, given by (19)

1 lnf κ r

(20)

(21)

ΔB = 0 Transitional regime

(15)

The shear-stress transport (SST) k-ω model developed by Menter to effectively blend the robust and accurate formulation of the k-ω model in the near-wall region with the freestream independence of the k-ε model in the far field. To achieve this, the k-ε model is converted into a k-ω formulation. The standard k-ω model and the transformed k-ε model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the nearwall region, which activates the standard k-ω model, and zero away from the surface, which activates the transformed k-ε model. The SST model incorporates a damped cross-diffusion derivative term in the ω equation. The definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress. These features make the SST k-ω model more accurate and reliable for a wider class of flows, for example, adverse pressure gradient flows, airfoils, transonic shock waves. The transport equtions for the SST k-ω turbulence model is given by

∂ ∂ ∂ ⎛ ∂k ⎞ (ρk ) + (ρkui ) = ⎜Γk ⎟ + Gk − Yk + Sk ∂t ∂x i ∂x j ⎝ ∂x j ⎠

(18)

where fr is a roughness function that quantifies the shift of the intercept due to roughness effects. Δ B depends on the shape and size of the roughness elements which varies from one type to the other making it nearly impossible to exactly simulate these roughness's. For uniform shaped roughness elements like those of a sand grain, Δ B correlates well with the non-dimensional roughness height, K+s = ρKsu∗/μ, where Ks is the physical roughness height. Formulas proposed by Cebeci and Bradshaw based on Nikuradses data are adopted to compute Δ B for each regime. Hydrodynamically smooth regime (K+s < 2.25):

⎟

2(γ − 1) ⎛ hlv ⎞ ⎛ hlv − 0.5⎞ (γ + 1) ⎝ RT ⎠ ⎝ RT ⎠

u*

1 1 Cμ4 k 2

ΔB =

And non-isothermal correction factor, theta, is given by

θ=

1 ⎛ ρu yp ⎞ ln E − ΔB κ ⎜ μ ⎟ ⎝ ⎠

and

2

2 ⎛ ρv ⎞ ⎛ 2σ ⎞ exp − ⎛ 4πr* σ ⎞ I= ⎟ ⎜ ⎟⎜ 3 (1 + θ) ⎝ ρl ⎠ ⎝ Mm π ⎠ ⎝ 3kB T ⎠ ⎜

=

where

(12)

where Td is the droplet temperature. The classical homogeneous nucleation theory assumes there are no impurities or foreign particles in the flow and predicts the formation of droplets from a supersaturated state. The nucleation rate described by the steady-state classical homogeneous nucleation theory and corrected for non-isothermal effects, is given by

qc

*

up u * τw / ρ

γ+1 ∂r P = Cp (Td − T ) ∂t hlv ρl 2πRT 2γ

(17)

ΔB =

+ ⎛ Ks

1 ln κ ⎝

⎜

(2.25
< 90):

− 2.25 + Cs Ks+ ⎞ × sin {0.4258(lnKs+ − 0.811)} 87.75 ⎠ ⎟

(22)

where Cs is a roughness constant, and depends on the type of the roughness. Fully rough regime (K+s > 90):

ΔB =

1 ln (1 + Cs Ks+) κ

(23)

Depending upon the roughness size and the type of flow regime a suitable Δ B is chosen from equations (21) and (22) or (23) and substituted back into equation (18) to calculate the wall properties like shear stress (τw ) for the mean temperature and turbulent quantities. The equations (1), (3) and (4) form a closed system of equations with equation (15) and allows for the calculation of the wet steam flow field. The roughness correction model used is based on the equivalent sand-grain approach. In the roughness correction model adopted here, sand grain height is introduced as a new parameter in the RANS turbulence model with wall functions to enhance turbulence in the wall region. This roughness model blends in perfectly with the Menters two equation SST k-w model. However, roughness models that are much

(16) 438

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

roughness correction model used in the present study clearly shows the influence of roughness on a turbulent boundary layer flow over a flat plate. It can hence be concluded from this study that the present roughness correction model is not influenced or contaminated by the consideration of non-equilibrium condensing wet steam flow in the current computational method. (ii) Pressure distribution along the centerline of the nozzle designed by Moses and Stein [1] for cases with different inlet condition are used in the present study. Two sets of pressure distribution data (case 410 and 421) along the center line of a converging-diverging nozzle with presumably smooth surface are considered here. The nozzle consists of two circular arcs meeting each other with a second degree continuity of curves, as seen in Fig. 2. The nozzle centerline is assigned a symmetric boundary condition and only half the nozzle is modelled. An expansion rate of 8230 s-1 results from a throat area of 10 mm × 10 mm. The data as made available by Moses and Stein [1] has been used to validate wet-steam flows in several studies and has been tested at higher pressures than the other nozzles available in the literature (ex.Moore nozzle [18]). As noticed from Fig. 5a and b a reasonably good agreement was found between the present numerical results and the experimental data [1].

more accurate are also present in the literature [43]. Nonetheless, the current roughness correction model performs quite precisely with acceptable amounts of error and is as shown in the validation section to follow the experimental values closely. 3. Computational domain The high expansion rate nozzle of Moses and Stein [1] is considered for the present study. The computational domain is two-dimensional in nature. The nozzle has converging and diverging parts having radii of 0.053 m and 0.686 m respectively meeting each other through a second order derivative continuity, as shown in Fig. 2. In order to capture the boundary layer a numerical grid of size 500×250 nodes was required for grid independent solutions. The nozzle was designed in such a way that the flow can be essentially treated as steady, one-dimensional, inviscid flow with attached supersonic expansion. A fine mesh at the nozzle walls ensured that the surface characteristics are properly captured and the desired y + value is achieved. As the mesh is fine near the walls, the aspect ratio is very high. Using a single-precision solver may produce erroneous convergence and accuracy due to inefficient transfer of boundary information and therefore the double-precision version of the Ansys Fluent 16 was used to obtain a high convergence value. Grid independence study is performed to reduce the influence of the grid size on the computational results. In the present study, the results presented were independent of the number and size of the mesh. Four different meshes were considered for this test, but the results of only the relevant three are presented here. The present grid is selected after it was coarsened and refined by decreasing and increasing the number of nodes by a factor of 2 in each direction. The table above shows the different grids and the number of control volumes each of it had along with the value of average y + at the walls. Non-dimensional pressure ratio was plotted against non-dimensional distance along the X-axis and compared for all the grids. The present grid and the finer mesh came closest to the actual experimental values and the residual RMS error values reduced to 10-5 with a mass imbalance of less than 2%. In all cases, steady-state solutions have been obtained on both present and refined meshes, which shows negligible difference and are therefore judged to be mesh independent. The details of the mesh independence study are tabulated in Table 1. Fig. 3 shows the graph of pressure variation along the nozzle centerline and how the solution changes with grid resolution. It is clearly seen from the graph that the currently chosen grid resolution is sufficient to obtain solutions close to the experimental values without an unnecessary increase in the computational time.

4.2. Pressure distributions Pressure distribution along the nozzle centreline was studied to observe any sudden jumps in local pressure values. Rapid condensation appears downstream of the throat owing to extremely high rate of expansion and a change in the radius of curvature of the wall of the nozzle. Due to the occurrence of nucleation, the droplets formed rapidly grow in size by exchanging heat and mass with the surrounding supercooled vapour. The high rate of heat release as a result of rapid condensation causes a sharp increase in vapour temperature. Depending on the values of flow parameters, the initial growth phase of droplets may give rise to a gradual increase in pressure known as condensation shock. The term shock is a misnomer, as although pressure rises as a result of heat addition to supersonic flow, the Mach numbers downstream of the condensation zone usually remain above unity and more importantly, the rise in pressure is gradual. This effect is well captured by the bumps in the pressure distribution plots which always occurs after the throat of the nozzle. After the condensation shock, the steam generally reverts to near thermodynamic equilibrium where the temperatures of both the vapour and the droplets are close to the saturation level. Since the growth of liquid phase takes place by heat transfer through finite temperature difference between the phases, the process is essentially irreversible and has associated with it net rise in entropy that appears as a reduction in the potential for performing work and is referred to as thermodynamic wetness loss. The back pressure in the present study is kept sufficiently low so that the flow is supersonic over at least some part of the diverging section. The variations of flow properties of the vapour phase can be explained approximately by considering the effects of change of area and external

4. Results and discussion 4.1. Validation The present numerical results are validated against two cases:(i) Transitionally and fully rough turbulent boundary layer cases with uniform spherical roughness elements having equivalent sand-grain roughness height of 0.79 mm are considered as studied by Ligrani & Moffat [44]. Free stream velocity of 10.1 m/s and 20.5 m/s were found to give a Reynolds number that causes transitionally rough behaviour on a flat surface. The dimensionless roughness height of 22.8 and 46.7 was obtained for these values of free stream velocities. The flat plate was of 5 m in length and the kinetic viscosity of the fluid was 1.5e-05 m2/s. The velocity profiles at an axial distance of 35.6% of the plate length were plotted. Velocity profiles given by the experiments conducted by Ligrani & Moffat are compared to the velocity profiles obtained as a result of using the current roughness correction model used in this paper. The predicted results of the simulation follow the experimental results very closely with a small and acceptable amount of error, as shown in Fig. 4. Therefore, the

Fig. 2. Geometry of the Moses and Stein Nozzle used for computation. 439

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

and 90 kPa, whereas the pressure at the outlet was maintained at 16 kPa and hence a pressure ratio of 4.5, 5 and 5.6 was maintained for the three cases studied. On increasing the surface roughness, it is observed that there is a slight rise in the pressure. The local pressure values are the least for smooth wall case, but as the roughness increases from 1 μm to a 1000 μm, the local pressure values also rise. However, in the region where the condensation shock appears, the phenomenon is exactly opposite, wherein, on subsequent increment of the surface roughness there is a fall in the local pressure value, thereby reducing the condensation shock strength. The drop in the peak pressure value of the condensation shock varies from 5% for 1 μm roughness to 9% for 1000 μm roughness for different inlet pressures. For the pressure variation plot of po = 80 kPa, the peak pressures of the condensation shock wanes on increasing the surface roughness values, as seen in Fig. 7. It can be observed that for po = 80 kPa, there is a second jump in pressure values at around 90% of the chord length. This is attributed to a secondary nucleation in the flow. The formation of new nuclei of droplets can occur via primary or secondary mechanisms. The former occurs directly from a supersaturated flow and the latter uses nuclei present in the flow as a site and source for the formation of new nuclei. The reduced energy required for secondary nuclei formation means that it can occur at lower supersaturation compared to primary nucleation. Primary nucleation is typically dominant only during the start-up phase of the process. Once a sufficient number of nucleus have formed and supersaturation drops within the metastable regime, secondary nucleation becomes the dominant mechanism for generation of new nuclei. This secondary nucleation again releases latent heat of formation into the flow raising the temperature and pressure of the flow and hence giving the pressure jump as seen the plot with po = 80 kPa. The back pressure in the present study is kept sufficiently low so that the flow is supersonic. If the inlet stagnation temperature is very high, then no appreciable nucleation takes place over the length of the nozzle and dry steam expands similar to the isentropic expansion of a perfect gas even though the steam might have become subcooled at the exit of the nozzle. At first Δ T increases continuously due to expansion. The steam becomes highly subcooled until the total surface area of the freshly nucleated droplets is sufficient to give rise to a significant rate of heat transfer. This causes the primary nucleation to take place and the droplets are formed and consequently grow in the flow. At this point, the liberated latent heat starts heating up the vapour and the subcooling drops very fast. Depending on the expansion rate, significant nucleation requires Δ T of the order of 30–40 K. In the present study, the given nozzle is expected to reach a subcooling of 40 K, as dictated by the flow properties of the vapour phase. But, if the subcooled vapour temperature exceeds this value, the expansion rate experiences a jump. The influence of expansion dominates that of heat addition and this suddenly increased expansion rate and consequently decreased Gibbs free energy facilitates another nucleation in the flow using nuclei already present in the flow as the site and source of formation of new droplet clusters. This form of nucleation is termed ”secondary nucleation” and is observed in the present study. The supersaturation drops for the flow with po = 80 kPa, and the maximum subcooled vapour temperature reaches above 50 K for the cases with Ks = 1 and 10 μm as can be seen from the plots in Fig. 9. It is seen that when the subcooled vapour temperature exceeds 45 K, secondary nucleation is bound to happen. This rise in the subcooled vapour temperature can be attributed to the fact that there is an abrupt jump in the expansion rate of the flow through the nozzle [45] [46] and it reaches a value as high as the first nucleation peak for these roughness values at po = 80 kPa, as seen in Fig. 10a and b, and hence this special flow feature is reserved only for these cases of flow and not for other rough flows with po = 70 or 90 kPa. It can be seen from the graphs in Fig. 11a and b that there is no wiggles in the expansion rate after the first peak, which shows primary nucleation, and hence the subcooled vapour temperature does not exceed 45 K and hence there is

Table 1 Details of grid independence study. Grid

Finer Mesh

Present Mesh

Coarser Mesh

Δ xmin Δ ymin No. of C.V.

0.061 m 0.7e-06 m 238,602 0.15

2.18e-04 m 1.4e-06 m 121,634 0.833

0.09869 m 2.8e-06 m 49,302 1.8

y+ ave

Fig. 3. Pressure variation plots changing with grid resolution.

Fig. 4. Velocity profiles for flat plate turbulent flow as given by experiments and present roughness correction model.

heat transfer. The position and structure of the condensation shock depend on the shape of the nozzle and the stagnation conditions, which together determine the flow conditions and the heat release rate due to condensation. It is seen that as the inlet pressure to the nozzle flow is increased the position of this condensation shock moves downstream as expected. The Fig. 6–8 show the effect of surface roughness on the pressure distribution and on the condensation shock for different inlet pressures by varying the surface roughness of the walls from hydraulically smooth to hydraulically rough regimes. The inlet pressures were set to 70.7, 80

440

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

Fig. 5. Validation of pressure variation along the centerline.

Fig. 6. Pressure Distribution at po = 70.7 kPa.

Fig. 7. Pressure Distribution at po = 80 kPa.

no secondary nucleation for po = 70 or 90 kPa. The expansion rate through the nozzle as approximated by Dykas et al. [47]. is given as

effects travel deeper in to the flow making its presence felt at greater distances. As an outcome of this effect the boundary-layer thickness increases and as the roughness values rise, the boundary layer thickness increases. This means that the shape parameter is higher for higher values of roughness, i.e., the boundary layer velocity profiles become less full and their resistance to separation due to shock dwindles. The boundary layer thickness is maximum for po = 80 kPa. This behaviour is owing to the phenomenon of secondary nucleation which is explained earlier. Typical values of boundary layer thickness as shown in Fig. 15 are 6.7× 10−4 , 10.8× 10−4 and 9.5× 10−4 corresponding to 70.7 kPa, 80 kPa and 90 KPa respectively.

c p − pout ⎞ P˙ = o ⎛ o po ⎝ c ⎠

(24)

4.3. Boundary layer velocity profiles & thickness The boundary-layer velocity distributions are very important parameters of a SWBLI. Shear forces play an important role as they neutralize the retardation imparted by the shock. The phenomenon is much more noticeable in those parts of the boundary-layer where the viscosity is quite low. The interaction mechanism depends on the thickness of the subsonic part of the boundary layer [32]. The variation in velocity profiles as the surface roughness is varied is seen in Fig. 12–14 at 70.7 kPa, 80 kPa and 90 kPa respectively. The position where the velocity profiles were drawn is selected based on the location of the peak of the condensation shock for the corresponding inlet pressure, which is at 67.5%, 71.6% and 76.67% for the respective inlet pressures in increasing order. As the surface roughness values increase, the viscous

4.4. Pressure gradient and skin friction coefficient During the interaction process the boundary-layer flow is retarded due to the presence of shock induced pressure gradient and also the shear stress. The retardation has a pronounced effect in the near wall region because of lower momentum. The resulting velocity profile is deeply influenced when the shear stress in the boundary-layer balances the abrupt pressure change by transferring momentum from the 441

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

external high velocity to internal low velocity regions. It is therefore pertinent to study the variation of the pressure gradient and skin friction coefficient on the boundary layer and observe how these values change in the presence of a rough wall surface. Pressure gradient rises in the vicinity of the condensation shock and the flow experiences an adverse pressure gradient. The continuous drop in the pressure gradient shows that the adverse pressure gradient gets weaker as the flow proceeds. Both the shear forces and adverse pressure gradient terms are reduced which results in a very weak interaction, one which does not lead to flow separation, although the flow always remained supersonic even after the shock. Separation however does occur in the flow towards the exit of the nozzle which does not have to do anything with the condensation SWBLI and is hence not discussed here. In Fig. 16, pressure gradient variation is shown for an inlet pressures of 90 kPa for different values of surface roughness. Fig. 16 shows the plot of pressure gradient when there is no shock and compares it with the one where shock does take place. It draws out the difference between the flows with and without condensation shock and the influence it has on the pressure gradient values. It is seen that there is a 50% reduction in the peak value of pressure gradient when the roughness increases from 1 μm to a 1000 μm. As the flow experiences a condensation shock, the pressure gradient values jump and raise as high as twice the value when there is no condensation shock. Also the variation of the pressure gradient along the nozzle axial direction is different for both the cases. The irregularity in the pressure gradient plot with wet steam and its difference from the case with dry air can be attributed to the jump in pressure due to condensation. While there is no shock in the case with dry air, the plot for pressure gradient is relatively flat. As the flow proceeds along the nozzle the Reynolds number of the flow increases and the boundary layer turns from laminar to turbulent. Due to this transition, the relative magnitude of the viscous forces decreases and hence the shear offered by the wall decreases. As condensation takes place, the shock causes a sudden and abrupt jump in the pressure thereby causing a jump in the local density value. The sudden jump in density and a reduction in the wall shear causes the skin friction to fluctuate. This is illustrated in the graphs of skin friction coefficient along the transonic wall in Fig. 17–19 where a drop is seen in the values at around 65–75% of the chord length. For an inlet pressure of 70.7 kPa, the drop is seen at approximately 65% of chord length, which increases to approximately 70% when the inlet pressure is increased to 80 kPa and 75% when the inlet pressure is further increased to 90 kPa. The secondary nucleation occurring at an inlet pressure of 80 kPa is very weak. The primary nucleation is relatively quite strong as evident

Fig. 8. Pressure Distribution at po = 90 kPa.

Fig. 9. Subcooled vapour temperature.

Fig. 10. Expansion rate through nozzle centerline. 442

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

Fig. 11. Expansion rate through nozzle centerline.

Fig. 12. Velocity Profile at po = 70.7 kPa at different roughness.

Fig. 14. Velocity Profile at po = 90 kPa at different roughness.

Fig. 13. Velocity Profile at po = 80 kPa at different roughness.

Fig. 15. Velocity profiles at different inlet pressures.

443

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

Fig. 16. Comparison of pressure gradient with and without shock at po = 90 kPa at different roughness. Fig. 19. Skin Friction Coefficient at po = 90 kPa.

from the dip that the graph of Cf takes. While the local skin friction value depends on the density and wall shear stress at that location, the ratio of change in wall shear stress to the change in density is what determines the value skin friction coefficient. This ratio remains approximately constant throughout the duration of the secondary nucleation and this results in a skin friction coefficient value that does not drop or fluctuate. This is shown in Fig. 7, where at approximately 90% of the chord length there is a secondary nucleation. In contrast, in Fig. 18, the skin friction coefficient remains constant approximately at 0.003. Therefore, as the condensation shock moves downstream on increasing the inlet pressure, so does the drop in skin friction coefficient. As the roughness of the walls increase, Cf values at a given pressure also increases. This is to be expected as there is an increment in the surface roughness values, the viscous effects are felt at a much larger distance from the wall and shear stresses are larger for larger roughness's. 5. Conclusion

Fig. 17. Skin Friction Coefficient at po = 70.7 kPa.

In the present study, the flow of wet steam following non-equilibrium condensation in the presence of wall surface roughness has been studied in a converging-diverging nozzle. The practical aim of the study in this field is to better understand the phenomenon associated with wet steam condensation for possible reduction of wetness losses in the last stages of steam turbines. A reduction in shock strength indicates a reduction in the entropy at outlet and hence the thermodynamic losses. It is observed that as the inlet pressure to the nozzle is increased the position of the condensation shock moves downstream. On increasing the surface roughness, a slight rise in the local pressure value is observed. However, in the region where the condensation shock appears, the phenomenon is exactly opposite, as on subsequent increment of the surface roughness there is a fall in the local pressure value. This indicates reduction of the condensation shock strength with increased roughness. The typical drop in the peak pressure value near the condensation shock varies from 5% for 1 μm roughness to 9% for 1000 μm roughness. As the Reynolds number increases, the shear forces become weak in comparison to the adverse pressure gradient imparted by the condensation shock. As the surface roughness values increase, the viscous effects affect deeper into the flow. As an outcome of this effect the boundary-layer thickness increases. Further, as the inlet pressure increases, the boundary-layer thickness increases. An average increase of about 33% was found in the boundary layer thickness in the presence of

Fig. 18. Skin Friction Coefficient at po = 80 kPa.

444

International Journal of Thermal Sciences 132 (2018) 435–445

A. Pillai, B.V.S.S.S. Prasad

condensation shock as compared to the case of dry air. Pressure gradient rises and is positive in the vicinity of the condensation shock, i.e., the flow experiences an adverse pressure gradient. It is seen that there is a 50% reduction in the peak value of pressure gradient when the wall roughness increases and it doubles as the flow experiences a condensation shock. This results in a very weak interaction, one which does not lead to flow separation. As condensation takes place, the shock causes a sudden and abrupt jump in the pressure thereby causing a jump in the local density value. The sudden jump in density and a reduction in the wall shear causes the skin friction to drop rapidly. For an inlet pressure of 70.7 kPa, the drop is seen at approximately 65% of chord length, which increases to approximately 70% when the inlet pressure is increased to 80 kPa and 75% when the inlet pressure is further increased to 90 kPa.

[18] M. Moore, P. Walters, R. Crane, B. Davidson, Predicting the fog drop size in wet steam turbines, Wet Steam 4 Conference, C37 1973 no. 73. [19] A. Binnie, M. Woods, The pressure distribution in a convergent-divergent steam nozzle, Proc. Inst. Mech. Eng. 138 (1) (1938) 229–266. [20] G. Gyarmathy, H. Meyer, Spontane kondensation teil 2: einfluss der entspannungsschnelligkeit auf die nebelbildung in übersättigtem dampf, VDIForschungsheft 508 (1965). [21] A. Townsend, The flow in a turbulent boundary layer after a change in surface roughness, J. Fluid Mech. 26 (2) (1966) 255–266. [22] R. Antonia, R. Luxton, The response of a turbulent boundary layer to a step change in surface roughness part 1. smooth to rough, J. Fluid Mech. 48 (4) (1971) 721–761. [23] X. Liu, L. Squire, An investigation of shock/boundary-layer interactions on curved surfaces at transonic speeds, J. Fluid Mech. 187 (1988) 467–486. [24] H. Babinsky, J. Edwards, Large-scale roughness influence on turbulent hypersonic boundary layers approaching compression corners, J. Spacecraft Rockets 34 (1) (1997) 70–75. [25] H. Babinsky, G. Inger, The effect of surface roughness on shock wave/turbulent boundary layer interactions, 18th AIAA Applied Aerodynamics Conference, Collection of Technical Papers, vol 18, AIAA, 2000, pp. 1–11. [26] H. Babinsky, G. Inger, Effect of surface roughness on unseparated shock-wave/ turbulent boundary-layer interactions, AIAA J. 40 (8) (2002) 1567–1573. [27] Y. Wu, K. Christensen, Spatial structure of a turbulent boundary layer with irregular surface roughness, J. Fluid Mech. 655 (2010) 380–418. [28] D. Squire, C. Morrill-Winter, N. Hutchins, M. Schultz, J. Klewicki, I. Marusic, Comparison of turbulent boundary layers over smooth and rough surfaces up to high Reynolds numbers, J. Fluid Mech. 795 (2016) 210–240. [29] A.A. Townsend, The Structure of Turbulent Shear Flow, Cambridge university press, 1980. [30] R.P. Taylor, Surface roughness measurements on gas turbine blades, ASME 1989 International Gas Turbine and Aeroengine Congress and Exposition, American Society of Mechanical Engineers, 1989V001T01A101–V001T01A101. [31] J.P. Bons, R.P. Taylor, S.T. McClain, R.B. Rivir, The many faces of turbine surface roughness, Trans. ASME-T-J. Turbomach. 123 (4) (2001) 739–748. [32] H. Babinsky, J.K. Harvey, Shock Wave-boundary-layer Interactions vol 32, Cambridge University Press, 2011. [33] J. Yuan, U. Piomelli, Numerical simulation of a spatially developing accelerating boundary layer over roughness, J. Fluid Mech. 780 (2015) 192–214. [34] R. Hanson, B. Ganapathisubramani, Development of turbulent boundary layers past a step change in wall roughness, J. Fluid Mech. 795 (2016) 494–523. [35] S.P. Schneider, Effects of roughness on hypersonic boundary-layer transition, J. Spacecraft Rockets 45 (2) (2008) 193–209. [36] Y. Zhou, Y. Zhao, D. Xu, Z. Chai, W. Liu, Numerical investigation of hypersonic flatplate boundary layer transition mechanism induced by different roughness shapes, Acta Astronaut. 127 (2016) 209–218. [37] A. White, Numerical investigation of condensing steam flow in boundary layers, Int. J. Heat Fluid Flow 21 (6) (2000) 727–734. [38] J. Starzmann, F.R. Hughes, A.J. White, M. Grübel, D.M. Vogt, Numerical investigation of boundary layers in wet steam nozzles, J. Eng. Gas Turbines Power 139 (1) (2017) 012606. [39] A. Nicol, J. Medwell, The effect of surface roughness on condensing steam, Can. J. Chem. Eng. 44 (3) (1966) 170–173. [40] P. Spalart, M. Shur, On the sensitization of turbulence models to rotation and curvature, Aero. Sci. Technol. 1 (5) (1997) 297–302. [41] G. Schnerr, P. Li, Shock wave/boundary layer interaction with heat addition by nonequilibrium phase transition, Int. J. Multiphas. Flow 19 (5) (1993) 737–749. [42] T. Setoguchi, S. Matsuo, S. Yu, H. Hirahara, Effect of nonequilibrium homogenous condensation on flow fields in a supersonic nozzle, J. Therm. Sci. 6 (2) (1997) 90–96. [43] L. Wang, Y. Zhao, S. Fu, Computational study of drag increase due to wall roughness for hypersonic flight, Aeronaut. J. 121 (1237) (2017) 395–415. [44] P.M. Ligrani, R.J. Moffat, Structure of transitionally rough and fully rough turbulent boundary layers, J. Fluid Mech. 162 (1986) 69–98. [45] H.A. Mashmoushy, On the Performance of Rotor Blades in Wet Steam, dissertation School of Manufacturing and Mechanical Engineering, University of Birmingham, 1994. [46] Z.A. Mamat, The Performance of a cascade of Nozzle Turbine Blading in Nucleating Steam, dissertation School of Manufacturing and Mechanical Engineering, University of Birmingham, 1996. [47] S. Dykas, M. Majkut, M. Strozik, K. Smołka, Losses estimation in transonic wet steam flow through linear blade cascade, J. Therm. Sci. 24 (2) (2015) 109–116.

References [1] C.A. Moses, G.D. Stein, On the growth of steam droplets formed in a laval nozzle using both static pressure and light scattering measurements, J. Fluid Eng. 100 (3) (1978) 311–322. [2] J. Joseph, S. Subramanian, K. Vigney, B. Prasad, D. Biswas, Thermodynamic wetness loss calculation in nozzle and turbine cascade: nucleating steam flow, Heat Mass Tran. (2017) 1–11. [3] P. Blythe, C. Shih, Condensation shocks in nozzle flows, J. Fluid Mech. 76 (3) (1976) 593–621. [4] M.R. Mahpeykar, A.R. Teymourtash, E.A. Rad, Theoretical investigation of effects of local cooling of a nozzle divergent section for controlling condensation shock in a supersonic two-phase flow of steam, Meccanica 48 (4) (2013) 815–827. [5] M.R. Mahpeykar, E. Amirirad, Suppression of condensation shock in wet steam flow by injection of water droplets in different regions of a laval nozzle, Sci. Iran. Trans. B Mech. Eng. 17 (5) (2010) 337. [6] K. Shimamoto, S. Matsuo, T. Setoguchi, K. Kaneko, Cfd investigation on passive control of condensation shock wave- in case of unsteady condensation shock, Trans. Jpn. Soc. Mech. Eng. B 67 (664) (2001) 135–142. [7] B. Haller, R. Unsworth, P. Walters, M. Lord, Wetness measurements in a model multistage low pressure steam turbine, Technology of Turbine Plant Operating with Wet Steam, 1989. [8] M. Moore, R. Jackson, N. Wood, R. Langford, P. Walters, K. Keeley, A method of measuring stage efficiency in operational wet steam turbines, Proc. Inst. Mech. Eng. C 179 (1979) 79. [9] F. Kreitmeier, W. Schlachter, J. Smutny, An investigation of flow in a low pressure wet steam model turbine and its use for determining wetness losses, Proc. Inst. Mech. Eng. 12 (1979) 385–395. [10] T. Tanuma, The Removal of Water from Steam Turbine Stationary Blades by Suction Slots 179 IMechEC423/022, 1991. [11] W. Steltz, P. Lee, W. Lindsay, The verification of concentrated impurities in lowpressure steam turbines, J. Eng. Power 105 (1) (1983) 192–198. [12] A. Kleitz, A. Laali, J. Courant, Fog droplet size measurement and calculation in wet steam turbines, Technology of Turbine Plant Operating with Wet Steam, 1989. [13] L. Fallick, M. Hopkinson, A. Jackson, Some aspects of low pressure steam turbine flow path design and analysis, Proceedings of IMechE Conference on Steam Turbines for the 1980s, Conference Publications, vol 12, 1979, pp. 253–280. [14] M. Stastny, Structure of secondary liquid phase downstream of moving blades in lp cylinder stages of steam turbine, Design Conference 1979 on Steam Turbines for the 1980s. Conference Sponsored by the Steam Plant Group and the Thermodynamics and Fluid Mechanics Group of the Institution of Mechanical Engineers, London, 912 October 1979, 1979. [15] F. Bakhtar, M.M. Tochai, An investigation of two-dimensional flows of nucleating and wet steam by the time-marching method, Int. J. Heat Fluid Flow 2 (1) (1980) 5–18. [16] P. Walters, Wetness and efficiency measurements in lp turbines with an optical probe as an aid to improving performance, J. Eng. Gas Turbines Power 109 (1) (1987) 85–91. [17] S. Dykas, M. Majkut, K. Smołka, M. Strozik, Study of the wet steam flow in the blade tip rotor linear blade cascade, Int. J. Heat Mass Tran. 120 (2018) 9–17.

445