Fill This Form To Receive Instant Help
Fill This Form To Receive Instant Help
Homework answers / question archive / 1)Examine how the rank of the stoichiometric matrix can be used to reduce the number of equations to be solved
1)Examine how the rank of the stoichiometric matrix can be used to reduce the number of equations to be solved. For example, if there are ns components and nr equations we can show that only ns − nr independent mass-balance equations are needed. The remaining variables form an invariant. Write MATLAB code to find the invariants.
2)Acetone is evaporating in a mixture of nitrogen and helium. Find the rate of evaporation and compare it with the rates in pure nitrogen and pure helium. Also compare it with the model using a pseudo-binary diffusivity value for acetone. The pseudo-binary diffusivity can be calculated from the correlation given by Wilke, Eq. (9.34).
3)The following data were obtained for the velocity profiles in turbulent flow at a specified point in a channel flow as a function of time. The measurements were taken 0.01 seconds apart. The velocity was reported in m/s. Find the mean velocity, the turbulent stress tensor (assume ρ = 1 kg/m3), and the kinetic energy of turbulence per unit mass.
4)Stability analysis with heat transfer. Set up the equations for steady state for the Bénard problem. Now perturb the temperature and use an energy equation to derive an equation for the temperature perturbation. This in turn needs an equation for the velocity profile. Derive this equation. Thus we have two perturbation variables to handle here. How would you proceed further?
5)Would the critical Rayleigh number for flow transition for the Bénard problem increase or decrease with the Prandtl number? Explain why in terms of the physics of the problem.The Pai equation for turbulent stress. Use the equation (17.19) suggested by Pai for the turbulent stress and integrate for the velocity profile. How do the results compare with that of Prandtl?
6)Stability of flow in torsional flow. Taylor determined the critical speed of rotation for flow between concentric cylinders with the inner cylinder rotating. The transition is characterized by a critical Taylor number, Ta, defined as Ta = R4 oκ(1 − κ)3(ω2 1 − ω2 2) ν2 The transition for only the inner cylinder rotating was obtained by Taylor as 1709 by performing a linear stability analysis. Set up a model and analyze this problem.
7)A shear layer between two fluids. Assume the following velocity distribution between two shear layers: vx(y) = v0 tanh(y/δ) This is known as the Betchov and Criminale form. Calculate the stability analysis using the Rayleigh equation for the following parameters: v0 = 1 and δ = 1. Set α = 0.8. Betchov and Criminale obtained cI = 0.1345 and cR = 0.
8)Kelvin–Helmholtz analysis. Derive the kinematic and dynamic conditions needed in the analysis. Set up the equations to find the constants. Now require that the determinant of the coefficient matrix should be zero to obtain a non-trivial solution. Hence verify the algebra leading to Eq. (16.32).
9)Inviscid-flow stability. Verify the integral obtained by Rayleigh and hence show that the velocity profile needs to have an inflexion point for instability. Show that a simple shear flow is stable. Hence viscosity is needed to cause flow instability for such cases.
10)There is a fluid evaporating from the surface. Here vx = 0, but vy = 0 at the plate surface. Derive the von Kármán momentum relation for this case.
11)Von Kármán assumed a cubic profile for the integral momentum analysis over a flat plate. Since a cubic has four constants, four conditions were used. (i) Vx = 0 at y = 0. (ii) Vx = Ve at y = δ. (iii) dVx/dy = 0 at y = δ. (iv) d2Vx/dy2 = 0 at y = 0. Show that the use of these conditions in a cubic profile leads to the representation given by Eq. (15.61). Justify condition (iv) above using the x-momentum balance applied at y = 0 together with the no-slip boundry condition.
12)Flow with an interfacial traction. Consider the problem of a semi-infinite fluid subject to a constant shear at the interface. This can be caused, for instance, by a surface-tension gradient. Show that the following differential equation is applicable for the boundary layer: 3f + 2ff − (f ) 2 = 0 State the boundary conditions. Use the program BVP4C to simulate the flow.
13)A cylinder of diameter 1.2 m and length 7.5 m rotates at 90 r.p.m. with its axis perpendicular to an air stream with an approach velocity of 3.6 m/s. Plot the tangential component of the velocity along the circumference of the cylinder. Plot the pressure distribution along the circumference. Find the lift force on the cylinder.
14)Verify the principle of superposition, which states that if φ1 and φ2 are solutions to potential flow then φ = c1φ1 + c2φ2 is also a solution. Also show that φ1φ2 is NOT a solution.Falkner–Skan flows. Find the value of m for which the wall shear stress is independent of the principal flow direction. Find the value of m for which the boundary-layer thickness is a constant. (The answer is m = 1.)
15)Consider again the problem of flow past a cylinder with rotation considered in Example 15.4. Calculate and plot the location of the stagnation point on the cylinder surface for 0 <>∗ < 4.="" here="">∗ is the dimensionless circulation defined as κ/(v∞R). Plot typical streamlines. Also calculate the locations for larger values of κ∗. These are now along the θ = −π/2 line but not on the surface of the cylinder.
