multi phase flow at high temperature in a vertical tube

multiphase flow

1. Numerical Methods for Modeling Phase Change Sandip kumar saha

2. Enthalpy Technique Energy equation (without melt convection): Now, H = h + Δ H Where H = Total enthalpy, h = Sensible heat, ∆H = Latent heat content Therefore Single domain formulation Extension to melt convection, complex interface, higher dimensions Where ΔH = for T < T solidus = f l L p for T solidus ≤ T ≤ T liquidus f l = liquid fraction = L p for T > T liquidus For a pure material, isothermal phase change occurs, i.e. T solidus = T liquidus = T m

3. Updating Nodal Heat Content Adding and subtracting appropriate terms to both sides When solution converges: Can may be rearranged as: (1) (2) n th iterative value a correction

4. Updating Nodal Heat Content Where n is the iteration number, f -1 is the inverse latent heat function For pure material, f -1 will be equal to cT m (2)-(1) Neglected and it is zero on convergence During phase change: Hence: Therefore:

5. Apparent Heat Capacity Method Thermodynamically, enthalpy = sensible + latent heats Specific heat is the rate of change of the enthalpy with respect to temperature : H of solid state heating, melting process and liquid state superheating: Therefore c p = c ps for T < T s = [ c psl +L/( T l – T s )] for T s ≥ T ≥ T l = c pl for T > T l Temperature field both in solid and liquid phase can be obtained by solving, T s T l c ps = c pl L/ Δ T c p (kJ/kg)

6. Multiphase Flow Modelling

7. Level Set Method A numerical technique for tracking evolution of interface, curve or surface . Involves use of a Level Set Function whose values define the interface or surface. Uses Eulerian approach - Interface movement is calculated on a fixed grid . Very useful methodology for modelling time-varying objects. Wide applications including two-phase flow, computer visualization, optimization etc.

8. Level Set Method: Concept Evolving of surface ϕ ( x,y,t ) with t x y z x y z x y z x y z ϕ = 0 z = 0 ϕ ( x,y,t ) = 0 ϕ = 0 z = 0 ϕ ( x,y,t ) = t ˊ ϕ = 0 z = 0 ϕ ( x,y,t ) = t ˊˊ ϕ = 0 z = 0 ϕ ( x,y,t ) = t ˊˊˊ where t ˊˊˊ> t ˊˊ > t ˊ

9. Level Set Function Level set function ϕ : Signed distance function . Each grid point in the domain is assigned a value of shortest distance from the interface ϕ = 0 : interface ϕ > 0 : phase 1. ϕ < 0 : phase 2 . In general, ϕ = c represents any general level (contour ). c represents a front which is defined to be all points where the surface has no height Interface ϕ = 0 Interface growth Osher and Sethian , J Comput . Phys., 1988 Sussman et. al., J Comput . Phys., 1994

10. Level Set Formulation Consider a point x = (x, y) on a front which evolves over time so that is its position x(t) over time. At any time t , for each point x(t) on the front the surface has by definition no height, therefore, ϕ ( x,y,t ) = 0 By chain rule Thus, → w here If n is normal to the front at x ˊ (t) and F is the speed function for that point , thus, x ′( t ) · n = F , where n = ∇ ϕ / |∇ ϕ | Hence evolution equation for ϕ is ϕ t + F | ∇ ϕ | =

11. Evolution of Level Set Function Level set equation is solved using advanced numerical techniques such as higher order finite difference schemes . Issues : Re-initialization and Mass correction – Mass is not implicitly conserved . After solving Level Set Equation , Volume fraction of a given phase calculated using Heavi -side function Volume averaging of properties done:

12. Volume of Fluid Method Numerical technique for tracking evolution of interface or free surface . Involves use of a Phase Fraction whose values define the two phases creating the interface. Unlike the level-set function, volume fraction of a phase is advected . Uses Eulerian approach - Interface movement is calculated on a fixed grid. Conserves mass of the moving phase. Used extensively for simulating free surface flows

13. Phase Fraction and its Advection Phase Fraction : f Advection equation : Solved using geometrical methods – Direct solution gives numerical error. Operator split technique required – Volume fraction updated in x and y directions separately . Properties averaged for new fraction values. Conserves mass and preserves shape v∆t u∆t Void phase Fluid v u U

14. Interface Reconstruction Interface reconstruction is required after advection – Maintains accurate interface shape in two-phase cells . Reconstruction done using Optimization Techniques : e.g. Least squares Volume-of fluid Interface Reconstruction algorithm (LVIRA). Minimizes the distance between the cell centre and interface location . Continuous iterations calculating interface normal and distance for a given volume fraction in two-phase cell. F =0 F =1 V i+1/2,j u i+1/2,j u i+1/2,j ∆t         liquid ñ gas l