Laminar boundary layer with hydrogen injection including multicomponent diffusion

Libby, Paul A.; Pierucci, Mauro

doi:10.2514/3.2752

Cited by 22 publications

(4 citation statements)

References 6 publications

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“…At equilibrium all the NH 3 and H 2 will react to form water as long as sufficient O 2 is available. Hence, at higher blowing rates the formation of H 2 O is assumed to occur within the boundary layer at a flame sheet, 17 where the flux of hydrogen atoms equals twice the flux of oxygen atoms. Additionally, 9 at the flame sheet has to be such that the heat transfer from that point matches the enthalpy of formation of the water reaction.…”

Section: Mathematical Modelmentioning

confidence: 99%

Dissociative cooling - Effect on stagnation heat transfer of gas mixture injection

Lacy

Wilson

Varghese

1995

Journal of Spacecraft and Rockets

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By embedding a dissociating material into the porous outer structure of a projectile, stagnation-point heat transfer may be reduced by transpiration cooling resulting from the outflow of the dissociated gas products. The principal material considered is ammonium chloride, NHLjCl, which dissociates at temperatures over 613 K, hence providing additional heat-transfer reduction. Stagnation-point heat-transfer solutions for the injection of the dissociation products of NH^Cl into both equilibrium and frozen boundary layers are presented and compared with those for the injection of other gases such as helium. Results show that dissociative cooling has the potential to provide a significant reduction in stagnation-point heat transfer as temperatures rise, because the gas injection rate increases with the temperature of the NH4C1 interface. NomenclatureCi = mass fraction, p//p c p = specific heat per unit mass at constant pressure c p = mass-average specific heat of gas mixture D -multicomponent diffusion coefficient T) = binary-diffusion coefficient / = quantity defined by Eq. (8) h = enthalpy 7 = diffusive mass flux k = thermal conductivity I = pn/(pn)u, Le = Lewis number, D p c p /k M -molar mass ___ M = blowing parameter, p w v w /^/p e^e K N = molar flux Nu = Nusselt number, q" w x/(T e ~ T w )k P = pressure Pr -Prandtl number, H,C P /K q" = heat flux R -gas constant r = cylindrical radius Re = Reynolds number, u e x/v T = temperature u = x -component velocity v = ^-component velocity x = distance along body surface y = distance normal to body surface s = well depth in intermolecular potential r) = transformation defined by Eq. (6) 9 = T/T e K = velocity gradient, du e /8x IJL = viscosity v = kinematic viscosity £ = transformation defined by Eq. (7) p = density a = collision diameter $>ij = function of molar masses, ^Mj/M t 0 = porosity

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Section: Mathematical Modelmentioning

confidence: 99%

Dissociative cooling - Effect on stagnation heat transfer of gas mixture injection

Lacy

Wilson

Varghese

1995

Journal of Spacecraft and Rockets

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“…= total number of points in the integration domain n p = point where G 0 n p 0 n Taylor = number of points for which the Taylor series expansion is applied Re = Reynolds number r = radial coordinate S = surface t = time U m = mean axial velocity u = velocity vector V = absolute injection velocity as seen by a stationary observer V = volume V w = injection velocity with respect to the moving wall z = axial coordinate α = wall expansion ratio β = scaled expansion ratio, α∕b γ i = ith coefficient in the Taylor series expansion ϵ = small real number used in numerical integration ε = small perturbation parameter η = radial transformation variable, 1∕2r 2 λ = scaling factor, 2∕Re ν = kinematic viscosity ξ = transformation variable, bη ξ max = domain integration size ξ p = coordinate location corresponding to n p ρ = density ψ = Stokes stream function Ω = vorticity vector Subscripts exit = denotes an exit or outlet section m = denotes a mean value max = denotes a maximum r, z = radial and axial component or partial derivative w = wall variable θ = tangential component 0 = fixed reference or initial value Superscript = dimensional variable I. Introduction V ISCOUS motion in cylindrical chambers with sidewall injection is of interest in a variety of applications, including mean flow modeling of solid [1][2][3] and hybrid rockets [4,5], sweat cooling [6,7], boundary-layer control [8][9][10], peristaltic pumping [11,12], gaseous diffusion, and isotope separation [13][14][15]. It is the latter group of studies by Berman [13,14] that has actually provided the impetus to develop the first similarity transformation of the Navier-Stokes equations into a fourth-order nonlinear ordinary differential equation (ODE).…”

Section: Doi: 102514/1j055949mentioning

confidence: 99%

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

Saad

Majdalani

2017

AIAA Journal

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The mean gaseous motion in solid rocket motors has been traditionally described using an inviscid solution in a porous tube of fixed radius and uniform wall injection. This model, usually referred to as the Taylor-Culick profile, consists of a rotational solution that captures the bulk gaseous motion in a frictionless rocket chamber. In practice, however, the port radius increases as the propellant burns, thus leading to time-dependent effects on the mean flow. This work considers the related problem in the context of viscous motion in a porous tube and allows the radius to be time dependent. By implementing a similarity transformation in space and time, the incompressible Navier-Stokes equations are first reduced to a nonlinear fourth-order ordinary differential equation with four boundary conditions. This equation is then solved both numerically and asymptotically, using the injection Reynolds number Re and the dimensionless wall regression ratio α as primary and secondary perturbation parameters. In this manner, closedform analytical solutions are obtained for both large and small Reynolds number with small-to-moderate α. The resulting approximations are then compared with the numerical solution obtained for an equivalent third-order ordinary differential equation in which both shooting and the irregular limit that affects the fourth-order formulation are circumvented. This code is found to be capable of producing the stable solutions for this problem over a wide range of Reynolds numbers and wall regression ratios.

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“…V iscous motion in cylindrical chambers with sidewall injection is of interest to a variety of applications including mean flow modeling of solid [1][2][3] and hybrid rockets, 4,5 sweat cooling, 6,7 boundary layer control, [8][9][10] peristaltic pumping, 11,12 gaseous diffusion, and isotope separation. [13][14][15] It is the latter group of studies by Berman 13,14 that has actually provided the impetus to develop the first similarity transformation of the Navier-Stokes equations into a fourth-order, nonlinear, ordinary differential equation (ODE).…”

mentioning

confidence: 99%

Viscous Flows Revisited in Simulated Rockets with Radially Regressing Walls

Saad¹,

Majdalani²

2011

47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference &Amp;amp; Exhibit

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The mean gaseous motion in solid and, less commonly, hybrid rocket motors has been traditionally described assuming inviscid flow in a porous cylinder of fixed radius and constant mass addition at the sidewall. This model, usually referred to as the Taylor-Culick profile, is a simple inviscid rotational solution that captures the bulk gaseous motion in a rocket chamber with sidewall injection. In practice, however, the radius of the rocket motor increases as the propellant burns, thus leading to time-dependent effects on the mean flow that are not embraced by the Taylor-Culick model. In this work, we revisit the problem of the viscous flow in a porous cylinder and allow the radius to be time dependent. By implementing Uchida's well established similarity transformation in space and time, the incompressible Navier-Stokes equations are first reduced to a nonlinear fourth-order ordinary differential equation with four boundary conditions that contain, in the case of an axisymmetric chamber, an irregular limit. This equation is then solved both numerically and asymptotically, using the injection Reynolds number, Re, and the dimensionless wall regression ratio, α, as primary and secondary perturbation parameters. At the outset, closed-form analytical solutions are obtained for both large and small Re with small-to-moderate α. The resulting approximations are then compared to the numerical solution obtained for an equivalent third-order ODE in which both shooting and the irregular limit are circumvented. We find our code capable of producing the stable solutions for this problem over a wide range of Reynolds numbers and wall regression ratios. The code also enables us to confirm the accuracy of the asymptotic approximations, both of which being presented either for the first time in the case of small suction and injection or reconstructed with additional detail in the case of large injection.

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Laminar boundary layer with hydrogen injection including multicomponent diffusion

Cited by 22 publications

References 6 publications

Dissociative cooling - Effect on stagnation heat transfer of gas mixture injection

Dissociative cooling - Effect on stagnation heat transfer of gas mixture injection

Viscous Mean Flow Approximations for Porous Tubes with Radially Regressing Walls

Viscous Flows Revisited in Simulated Rockets with Radially Regressing Walls

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