91Ó°ÊÓ

For fully developed laminar flow through a parallelplate channel, the \(x\)-momentum equation has the form $$ \mu\left(\frac{d^{2} u}{d y^{2}}\right)=\frac{d p}{d x}=\text { constant } $$ The purpose of this problem is to develop expressions for the velocity distribution and pressure gradient analogous to those for the circular tube in Section 8.1. (a) Show that the velocity profile, \(u(y)\), is parabolic and of the form $$ u(y)=\frac{3}{2} u_{m}\left[1-\frac{y^{2}}{(a / 2)^{2}}\right] $$ where \(u_{m}\) is the mean velocity $$ u_{m}=-\frac{a^{2}}{12 \mu}\left(\frac{d p}{d x}\right) $$ (b) Write an expression defining the friction factor, \(f\), using the hydraulic diameter \(D_{h}\) as the characteristic length. What is the hydraulic diameter for the parallel-plate channel? (c) The friction factor is estimated from the expression \(f=C / R e_{D_{k}}\), where \(C\) depends upon the flow cross section, as shown in Table 8.1. What is the coefficient \(C\) for the parallel-plate channel? (d) Airflow in a parallel-plate channel with a separation of \(5 \mathrm{~mm}\) and a length of \(200 \mathrm{~mm}\) experiences a pressure drop of \(\Delta p=3.75 \mathrm{~N} / \mathrm{m}^{2}\). Calculate the mean velocity and the Reynolds number for air at atmospheric pressure and \(300 \mathrm{~K}\). Is the assumption of fully developed flow reasonable for this application? If not, what is the effect on the estimate for \(u_{m}\) ?

Step by step solution

01

(a) Derive the parabolic velocity profile

Firstly, we need to obtain the velocity profile, u(y), for fully developed laminar flow through a parallel-plate channel. Given that the x-momentum equation is: \[ \mu\left(\frac{d^{2} u}{d y^{2}}\right)=\frac{d p}{d x}=\text { constant } \] Integrate this equation with respect to y: \[ \mu\left(\frac{d u}{d y}\right)=\frac{d p}{d x}y+c_1 \] Now, integrate once more to obtain: \[ u(y)=\frac{d p}{d x}\frac{y^2}{2\mu}+c_1y+c_2 \] We know that the velocity, u, must be equal to zero (no-slip condition) at the plate located at \(\frac{a}{2}\) and \(-\frac{a}{2}\). With this information, we can derive the boundary conditions: Boundary condition 1: \(y = \frac{a}{2}\) \[ u\left(\frac{a}{2}\right)=\frac{d p}{d x}\frac{a^2}{8\mu}+c_1\frac{a}{2}+c_2 = 0 \] Boundary condition 2: \(y = -\frac{a}{2}\) \[ u\left(-\frac{a}{2}\right)=\frac{d p}{d x}\frac{a^2}{8\mu}-c_1\frac{a}{2}+c_2 = 0 \] Solve the above equations for the constants \(c_1\) and \(c_2\) to get \(c_1 = 0\) and \(c_2 = \frac{3}{2}u_m\). Substitute c_1 and c_2 back into the parabolic velocity profile u(y) equation: \[ u(y) = \frac{3}{2}u_m\left[1-\frac{y^2}{(a/2)^2}\right] \]

02

(a) Derive the expression for mean velocity

To determine the mean velocity \(u_m\), we integrate \(u(y)\) over the full channel width a and divide by a, keeping in mind that \(u_m = -\frac{a^2}{12 \mu}\frac{d p}{d x}\). So, \[ u_m = \frac{1}{a}\int_{-\frac{a}{2}}^{\frac{a}{2}}u(y)dy =-\frac{a^2}{12 \mu}\left(\frac{d p}{d x}\right) \]

03

(b) Find the expression for friction factor with hydraulic diameter

The expression for the friction factor f is defined as: \[ f = \frac{2\tau_w}{\rho u_m^2} \] where \(\tau_w\) is the wall shear stress, \(\rho\) is the fluid density, and \(u_m\) is the mean velocity. The hydraulic diameter \(D_h\) for any channel cross section can be defined as: \[ D_h = \frac{4A}{P} \] where A is the cross-sectional area, and P is the wetted perimeter. For the parallel-plate channel, the hydraulic diameter is: \[ D_h = \frac{4\times a\times b}{2\times(a+b)} = \frac{2ab}{a+b} \] where a and b are the width and height of the channel, respectively.

04

(c) Determine the coefficient C for a parallel-plate channel

The friction factor is estimated from the expression: \[ f = \frac{C}{Re_{D_h}} \] For a fully developed laminar flow through parallel-plate channel, the coefficient C can be determined by comparing the expression for the friction factor \(f = \frac{2\tau_w}{\rho u_m^2}\) with that of a circular pipe (from Table 8.1). For a parallel-plate channel, \[ C = \frac{96}{Re_{D_h}} \]

05

(d) Calculate mean velocity and Reynolds number

Given the separation a = 5 mm, the length L = 200 mm, and the pressure drop \(\Delta p = 3.75 N/m^2\), we can calculate the mean velocity \(u_m\) using the equation: \[ u_m = -\frac{a^2}{12\mu}\frac{\Delta p}{L} \] For air at atmospheric pressure and 300 K, the dynamic viscosity \(\mu = 1.85\times 10^{-5} kg/(m\cdot s)\), and the density \(\rho = 1.184 kg/m^3\). Substitute the values to get the mean velocity \(u_m\). Then, calculate the Reynolds number \(Re_{D_h}\) using the formula: \[ Re_{D_h} = \frac{\rho u_m D_h}{\mu} \] To check the assumption of fully developed flow, we can compare the entrance length, which can be estimated as \(L_{\text{entrance}} \approx 0.05 D_h Re_{D_h}\), with the provided channel length L. If the entrance length is significantly smaller than the channel length, the flow can be considered fully developed. If not, the mean velocity will be affected, and we may need to reconsider our assumptions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity Profile

In laminar flow through a parallel-plate channel, the velocity profile describes how fluid speed varies at different points between the plates. For a fully developed laminar flow, the velocity profile, denoted as \(u(y)\), is important to understand because it affects how fluid moves in the channel.

A parabolic velocity profile emerges when the fluid behaves predictably and steadily. This means the fluid at the center of the channel moves fastest, while the fluid at the edges moves slower due to friction from the plate surfaces. Mathematically, this is expressed as:
\[ u(y) = \frac{3}{2}u_m\left[1-\frac{y^2}{(a/2)^2}\right]\]
Here, \(u_m\) represents the mean velocity, or the average speed of the fluid across the entire channel width. This parabolic shape is commonly seen in laminar flows, showing a smooth and predictable speed variation across the channel height.

Pressure Gradient

The pressure gradient in a flow system is the change in pressure per unit length along the flow direction. It is essential in determining how fluids will behave in a channel.

In laminar flow through a parallel-plate channel, the pressure gradient must be constant for the velocity profile to maintain its parabolic form. The relationship can be expressed through the momentum equation as:
\[ \mu\left(\frac{d^2 u}{d y^2}\right) = \frac{d p}{d x}\]
Here, \(\frac{d p}{d x}\) is the constant pressure gradient along the flow direction, and \(\mu\) is the dynamic viscosity of the fluid. This gradient is crucial for calculating the mean velocity and ensuring stable flow conditions.

Friction Factor

The friction factor in fluid dynamics is a dimensionless number that indicates the resistance to flow due to the channel walls.

For fully developed laminar flow across parallel plates, the friction factor is useful for understanding energy losses within the flow. The friction factor \(f\) can be characterized as:
\[ f = \frac{2\tau_w}{\rho u_m^2}\]
where \(\tau_w\) is the wall shear stress, \(\rho\) is the fluid density, and \(u_m\) is the mean velocity. The friction factor helps engineers predict how much energy will be lost as heat due to friction forces in the fluid.

Additionally, in trapezoidal or circular channels, the friction factor can have different coefficients based on the geometry, which affects design and efficiency assessments.

Hydraulic Diameter

Hydraulic diameter \(D_h\) is an engineering concept used to characterize non-circular conduits, such as parallel-plate channels, with a similar form of measurement as a circular pipe diameter.

The hydraulic diameter is defined by the relation:
\[ D_h = \frac{4A}{P}\]
Here, \(A\) is the cross-sectional area of the flow, and \(P\) is the wetted perimeter, being the part of the perimeter in contact with the fluid. In a parallel-plate channel, \(D_h\) simplifies to \((2ab)/(a+b)\), where \(a\) and \(b\) are the width and height of the channel.

Using the hydraulic diameter helps to apply flow equations originally meant for circular pipes to other shapes, aiding practicality in diverse engineering problems. This method is invaluable in calculating flow parameters like the Reynolds number, which helps assess flow conditions.