Page 2 Of 43: The Uniform Stream Of Velocity U Approaches

Page 2 Of 43 A The Uniform Stream Of Velocity U Approaches A Channe

Show that the entrance length le is proportional to the Reynolds number Re (based on the channel width b). Fluid’s dynamic viscosity is ï and density is ï². (10 points) The uniform stream of velocity Uï‚¥ and temperature Tï‚¥ approaches stationary flat plate whose temperature is TW (TW > Tï‚¥). Fluid’s kinematic viscosity is ï®, thermal diffusivity is ï«, and Prandtl number Pr = 10. Estimate distances xv and xT (measured from the leading edge) where velocity boundary layer reaches thickness ï¤v ~ L and thermal boundary layer reaches thickness ï¤T ~ L, respectively. What is the ratio xT / xv ? (15 points) Obtain an expression for the flow rate Q as a function of fluid viscosity (ï), pressure difference (pA - pB), tube radius (R), and tube length (L). Consider laminar flow through straight circular pipe, neglecting elbow effects and body forces, and noting the mean velocity is half of maximum velocity.

Paper For Above instruction

The analysis of entrance length in laminar flow within a channel is fundamental in fluid mechanics, revealing how flow develops from initial uniform conditions to fully developed flow conditions. The length, termed the entrance length (le), is directly proportional to the Reynolds number (Re), which characterizes the ratio of inertial to viscous forces in the flow, and is based on the characteristic dimension, in this case, the channel width b.

The Reynolds number in a channel flow is defined as:

Re = (U * b) / ν

where U is the flow velocity, b is the channel width, and ν is the fluid’s kinematic viscosity. The development of the boundary layer from the inlet to the point where the flow becomes fully developed involves viscous diffusion across the flow. Empirical and theoretical studies have shown that the entrance length (le) in laminar flow scales linearly with the Reynolds number:

le ≈ 0.05 Re b

indicating that the length needed for the velocity profile to stabilize is proportional to Re and the channel width. This proportionality stems from the balance wherein higher flow velocities or larger characteristic lengths promote longer development regions, moderated by viscous effects.

In the thermal boundary layer scenario, the characteristics of the velocity and thermal boundary layers are governed by similar diffusion processes but with different diffusivities. For a flat plate with free-stream velocity Uï‚¥ and temperature Tï‚¥ approaching a stationary plate at temperature TW, the boundary layers develop over distance from the leading edge. The thermal boundary layer thickness ï¤T and velocity boundary layer thickness ï¤v are estimated by similarity solutions:

ï¤v ≈ 5 (x ν / U)^{1/2}

ï¤T ≈ 5 (x α / U)^{1/2}

where x is the distance from the leading edge, α is thermal diffusivity, and Pr = ν / α = 10. To find positions xv and xT where these boundary layers reach specific characteristic thickness L, we solve for x:

xv ≈ (L / 5)^2 * (U / ν)

xT ≈ (L / 5)^2 * (U / α)

The ratio of these distances:

xT / xv = (α / ν) = Pr = 10

which confirms that the thermal boundary layer extends further than the velocity boundary layer in a Prandtl number greater than 1.

For flow through a circular pipe, the volumetric flow rate Q can be derived from the Hagen-Poiseuille law for laminar flow:

Q = (π R^4 / 8 μ) * (pA - pB) / L

where μ is the dynamic viscosity, pA - pB is the pressure difference between the pipe ends, R is the radius, and L is the length of the pipe. The mean velocity V̄ is related to the maximum velocity Vm (which occurs at the center) by V̄ = Vm / 2. Substituting this into the flow rate equation and recognizing that the pressure gradient is related to the pressure difference gives:

Q = π R^2 V̄ Area = π R^2 (Vm / 2) * (π R^2)

leading to the classical Hagen-Poiseuille equation after rearrangement:

Q = (π R^4 / 8 μ) * (pA - pB) / L

This formula allows calculation of the flow rate given the fluid's viscosity, the pressure difference, the radius, and the length, assuming laminar flow and neglecting other effects such as elbows or body forces.

References

  • White, F. M. (2011). Fluid Mechanics (7th ed.). McGraw-Hill Education.
  • Sharma, A., & Verma, S. K. (2018). Fundamentals of Fluid Mechanics. Springer.
  • Munson, B. R., Young, D. F., & Okiishi, T. H. (2013). Fundamentals of Fluid Mechanics (7th ed.). Wiley.
  • Çengel, Y. A., & Ghajar, A. J. (2015). Heat and Mass Transfer (5th ed.). McGraw-Hill Education.
  • Kundu, P. K., Cohen, I. M., & Dowling, D. R. (2015). Fluid Mechanics (6th ed.). Elsevier.
  • Fox, R. W., McDonald, A. T., & Pritchard, P. J. (2011). Introduction to Fluid Mechanics (8th ed.). Wiley.
  • Cengel, Y. A., & Boles, M. A. (2014). Thermodynamics: An Engineering Approach (8th ed.). McGraw-Hill Education.
  • Rao, Y. V. C., & Reddy, P. V. (2014). Heat Transfer. New Age International.
  • Batchelor, G. K. (2000). An Introduction to Fluid Dynamics. Cambridge University Press.
  • Launder, B. E., & Spalding, D. B. (1974). The Numerical Simulation of Turbulent Flows. Computer Methods in Applied Mechanics and Engineering.

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