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Mechanical Engineering

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    Introduction

    In previous reference material, governing principles of fluid mechanics were developed using conservation of mass, momentum, and energy. Many practical engineering systems apply these principles to flows moving through enclosed passages such as pipes, tubes, ducts, and channels.

    Pipe flow is important in a wide range of applications including water distribution systems, HVAC systems, hydraulic machinery, oil and gas transport,cooling systems, and more. Although the geometry of a pipe may appear simple, the resulting flow behavior can vary significantly depending on the effects of viscosity, pressure gradients, pipe roughness, and flow speed.

    Unlike certain inviscid flow models used in earlier chapters, viscous effects in pipe flow are often dominant and directly influence the pressure losses required to maintain fluid motion. Energy supplied by pumps or elevation differences is continually dissipated through viscous shear within the fluid and at the pipe walls. This chapter focuses on the behavior of incompressible viscous flow in pipes. Important topics include laminar and turbulent flow behavior, entrance region development, pressure losses, velocity profiles, friction factors, and pipe system analysis. Both major losses associated with wall friction and minor losses associated with components such as valves, bends, and fittings are considered as well.

    Engineers will find this information useful in analyzing the performance of pipe systems as well as sizing components like pumps or air conditioners to achieve a particular thermo-fluidic output.

    Basic Theory of Pipe Flow

    Pipe flow refers to fluid motion through enclosed passages that are completely filled with fluid. This is contrasted with what is commonly termed open-channel flow in which the fluid conduit is only partially occupied with fluid with the rest being exposed to ambient conditions such as in a gutter. It should be noted that the mechanics of open-channel flow are different from pipe flow.

    An important and often defining feature of pipe flow is that fluid motion is usually driven by a pressure difference or gradient along a length of pipe. As the fluid moves downstream, viscous effects at the wall create shear stresses that resist the motion of the fluid. This resistance produces pressure losses and converts mechanical energy into thermal energy through viscous dissipation. In this way pipe flow represents an interaction between pressure gradients and viscous forces that define the velocity profile and flow characteristics inside of the pipe conduit.

    Laminar and Turbulent Flow

    Pipe flow may be classified as laminar, transitional, or turbulent. For laminar flow, fluid particles move in relatively smooth layers with limited mixing between neighboring fluid elements. In turbulent flow, velocity and pressure fluctuate continuously and strong mixing occurs throughout the flow. Transitional flow occupies an intermediate state, where smooth flow characeristics are unstable or sporadically interrupted by turbulent mixing. Perhaps unsurprisingly, whether a flow displays laminar or turbulent characteristics depends primarily on the Reynolds number. Indeed, viscous forces are dominant in laminar flow whereas inertial forces are more significant in turbulent flow. For pipe flow, the Reynolds number is given as:
    $$ Re=\frac{\rho V D}{\mu} $$
    where:
    • \( \rho \) is fluid density,
    • \( V \) is average velocity,
    • \( D \) is pipe diameter,
    • \( \mu \) is dynamic viscosity.
    It should be noted that transition from laminar to turbulent flow is not governed strictly by the Reynolds number and thus precise cutoff values can't be established. Nevertheless, rules of thumb or practical ranges for circular pipe flow can still be used as follows:
    $$ Re \lesssim 2300 \qquad \text{Laminar} $$
    $$ 2300 \lesssim Re \lesssim 4000 \qquad \text{Transition} $$
    $$ Re \gtrsim 4000 \qquad \text{Turbulent} $$
    Visualization of laminar and turbulent pipe flow

    Defining fully developed flow

    When fluid first enters a pipe, the velocity profile is often nearly uniform. Due to the no-slip condition, fluid near the wall slows down and boundary layers (boundary layers are covered in greater detail in Chapter 9; in pipe flow this is a region where viscous effects are relevant to the flow) begin growing inward from the pipe surface.

    As the flow continues downstream, the boundary layers increase in thickness until they merge at the pipe centerline. Beyond this point, the velocity profile no longer changes shape in the flow direction and the flow is said to be fully developed. The distance required for the flow to become fully developed is called the entrance length.The final shape of the flow profile as well as the entrance length depends on the nature of the incoming flow, be it laminar or turbulent. For laminar flow, the entrance length is approximately:

    $$ \frac{L_e}{D}\approx 0.06Re $$
    while turbulent flow generally develops more rapidly
    $$ \frac{L_e}{D}\approx 4.4Re^{1/6} $$

    Pressure gradients: The drivers of pipe flow

    For fully developed flow in a horizontal pipe, pressure decreases in the flow direction because energy is continually lost due to viscous shear at the wall:
    $$ \frac{dp}{dx}<0 $$
    The wall shear stress provides the resisting force that balances the pressure force driving the flow. In fully developed flow, the pressure gradient is approximately constant along the pipe:
    $$ \frac{dp}{dx}=\text{constant} $$
    The relationship between pressure losses and wall shear stresses forms the basis for analyzing practical pipe flow systems. Note that depending on the orientation of the pipe, gravity may operate as a driving force, supporting the pressure gradient or a countervailing force, working against the pressure gradient alongside viscosity.
    Visualization of laminar and turbulent pipe flow

    Developed Laminar Flow

    For sufficiently small Reynolds numbers, pipe flow remains laminar and fluid motion occurs in relatively smooth layers with limited mixing between adjacent fluid particles. After the entrance region ends, the velocity profile no longer changes shape in the flow direction and the flow becomes fully developed.

    For steady, incompressible, fully developed flow in a horizontal circular pipe, the axial velocity depends only on radial position:

    $$ u=u(r) $$
    The radial and circumferential velocity components are negligible:
    $$ v_r=0, \qquad v_\theta=0 $$
    and pressure varies only in the axial direction:
    $$ p=p(x) $$
    Because the flow is fully developed, pressure decreases linearly along the pipe length. A pressure difference is therefore required to overcome viscous resistance and maintain fluid motion

    Velocity Distribution

    Applying the momentum equation together with Newton's law of viscosity leads to the classical laminar pipe velocity profile:
    $$ u(r)= \frac{\Delta p}{4\mu L} \left( R^2-r^2 \right) $$
    where:
    • \( R \) is the pipe radius,
    • \( r \) is the radial coordinate,
    • \( \Delta p \) is the pressure drop across pipe length \( L \).
    The profile is parabolic, with the maximum velocity occurring at the centerline:
    $$ u_{\max} = \frac{\Delta pR^2}{4\mu L} $$
    The no-slip condition requires:
    $$ u(R)=0 $$
    while symmetry gives:
    $$ \left( \frac{du}{dr} \right)_{r=0}=0 $$

    Pressure Drop and Wall Shear Stress

    Take note from the preceding equations that the pressure drop required to maintain flow increases with:
    • increasing flow rate,
    • increasing viscosity,
    • increasing pipe length,
    and decreases with increasing pipe diameter. The wall shear stress is related to the velocity gradient at the wall:
    $$ \tau_w = \mu \left( \frac{du}{dr} \right)_{r=R} $$
    For fully developed laminar pipe flow:
    $$ \tau_w = \frac{\Delta pD}{4L} $$
    showing that the wall shear stress is directly related to the pressure loss along the pipe.

    Darcy Friction Factor

    Pressure losses in pipe flow are commonly modeled using the Darcy friction factor:
    $$ \Delta p = f\frac{L}{D} \frac{\rho V^2}{2} $$
    The friction factor is a dimensionless quantity that characterizes the amount of resistance or energy loss produced by wall friction within the pipe. For fully developed laminar flow in a circular pipe:
    $$ f=\frac{64}{Re} $$
    Unlike turbulent flow, the laminar friction factor depends only on Reynolds number and is independent of pipe roughness.

    Fully Developed Turbulent Pipe Flow

    At sufficiently large Reynolds numbers, pipe flow becomes turbulent and the fluid motion contains strong velocity fluctuations and mixing throughout the flow. Unlike laminar flow, turbulent motion continuously transfers momentum between different fluid regions, producing larger wall shear stresses and greater pressure losses.

    As with laminar flow, turbulent pipe flow eventually becomes fully developed after the entrance region. In fully developed turbulent flow, the average velocity profile no longer changes shape in the flow direction, although instantaneous velocity fluctuations still occur.

    Velocity Profile

    Turbulent mixing causes momentum to be transported more effectively across the pipe cross section than in laminar flow. As a result, the turbulent velocity profile is generally flatter through most of the pipe interior with steep velocity gradients near the wall.

    Unlike laminar flow, there is no simple exact analytical solution for the full turbulent velocity distribution. However, approximate empirical profiles are commonly used. One common approximation is the power-law profile:

    $$ \frac{u}{u_{\max}} = \left( 1-\frac{r}{R} \right)^{1/n} $$
    where:
    • \( u \) is the local velocity,
    • \( u_{\max} \) is the centerline velocity,
    • \( R \) is the pipe radius,
    • \( n \) is an experimentally determined exponent.
    For many turbulent pipe flows:
    $$ n\approx 7 $$
    giving the so-called one-seventh power law.

    Pressure Losses

    Pressure losses in turbulent pipe flow are substantially larger than in laminar flow because of increased momentum mixing and wall shear stress. The pressure drop is commonly expressed using the Darcy--Weisbach equation:
    $$ \Delta p = f\frac{L}{D} \frac{\rho V^2}{2} $$
    where:
    • \( f \) is the Darcy friction factor,
    • \( L \) is pipe length,
    • \( D \) is pipe diameter,
    • \( V \) is average velocity.

    Friction Factor Behavior

    Unlike laminar flow, the turbulent friction factor depends on both Reynolds number and pipe roughness:
    $$ f=f(Re,\epsilon/D) $$
    where:
    $$ \epsilon $$
    is the characteristic roughness height. For smooth pipes at moderate Reynolds numbers, empirical correlations such as the Blasius equation are often used such as:
    $$ f\approx 0.316Re^{-1/4} $$
    More general turbulent pipe flow calculations commonly use the Moody chart or the Colebrook equation which will be discussed below in a broader discussion about pipe losses. Most empirical formulae such as the one presented above are just approximations of the Colebrook equation for certain conditions.

    Energy Dissipation in Turbulent Flow

    Turbulent pipe flow produces greater energy dissipation than laminar flow because eddies and continuous chatoic swirling continually transfers momentum and kinetic energy throughout the fluid, prompting greater losses due to friction or viscosity. As Reynolds number increases, turbulent mixing generally becomes stronger, leading to larger pressure losses and usually greater pumping power requirements.

    Pipe Flow Losses

    As fluid moves through a pipe system, mechanical energy is continually lost due to viscous effects. These losses appear primarily as pressure drops and must often be overcome using pumps, fans, or elevation differences. Pipe flow losses are commonly divided into two categories:
    • major losses,
    • minor losses.
    Broadly speaking, major losses encompass energy dissipation due to frictional effects between the working fluid and the pipe wall. Minor losses result when a flow is forced to alter its speed, direction or flow geometry usually due to pipe fitting or bend in the flow. These losses result from the local formation of turbulence and eddy formulation that enhances energy dissipation. While major losses are usually the more significant culprit of pressure drop in a system, minor losses need not be small in magnitude.

    Major Losses

    Major losses are associated with viscous friction along the walls of a pipe. These losses accumulate continuously over the pipe length and are usually the dominant source of pressure drop in long pipe systems.

    The pressure loss due to wall friction is commonly modeled using the Darcy--Weisbach equation:

    $$ \Delta p = f\frac{L}{D} \frac{\rho V^2}{2} $$
    or, expressed as head loss:
    $$ h_f = f\frac{L}{D} \frac{V^2}{2g} $$
    where:
    • \( f \) is the Darcy friction factor,
    • \( L \) is pipe length,
    • \( D \) is pipe diameter,
    • \( V \) is average velocity.
    For laminar flow, the friction factor is given by the analytical expression given in the discussion on developed laminar flow, while turbulent flow requires empirical correlations or the Moody chart since the friction factor depends on both Reynolds number and pipe roughness.
    Visualization of major losses and the need for a driving pressure differential in pipe flow

    Moody Chart

    As already stated above, for turbulent pipe flow, the Darcy friction factor depends on both Reynolds number and relative roughness:
    $$ \frac{\epsilon}{D} $$
    where:
    • \( \epsilon \) is the average roughness height,
    • \( D \) is the pipe diameter.
    The Moody chart provides a graphical method for determining the friction factor:
    $$ f=f\left(Re,\frac{\epsilon}{D}\right) $$
    Moody Chart for calculating major losses. Used from Wikipedia under the Creative Commons Attribution-Share Alike 4.0 International license
    Moody Diagram Source: (https://en.wikipedia.org/wiki/Moody_chart#) To use the Moody chart:
    1. Compute the Reynolds number:
      2. $$ Re=\frac{\rho VD}{\mu} $$
    2. Determine the pipe relative roughness:
      3. $$ \frac{\epsilon}{D} $$
    3. Locate the Reynolds number on the horizontal axis.
    4. Move vertically until intersecting the appropriate relative
      5. roughness curve.
    5. Read the Darcy friction factor from the vertical axis.
    6. The friction factor can then be used to estimate major losses in a pipe system
    Note again that laminar flow does not depend on relative roughness and thus only a single analytical curve is shown for completeness on a typical Moody diagram. In turbulent flow, increasing roughness generally increases the friction factor and therefore increases pressure losses within the pipe system.

    Colebrook Equation

    When working with the Moody diagram and major losses, an engineer might hear of the Colebrook formula. The Colebrook equation provides an implicit relationship for the Darcy friction factor in turbulent pipe flow:
    $$ \frac{1}{\sqrt{f}} = -2\log_{10} \left( \frac{\epsilon}{3.7D} + \frac{2.51}{Re\sqrt{f}} \right) $$
    where:
    • \( f \) is the Darcy friction factor,
    • \( Re \) is the Reynolds number,
    • \( \epsilon \) is the pipe roughness,
    • \( D \) is the pipe diameter.
    Because \( f \) appears on both sides of the equation, the Colebrook equation must be solved iteratively. The Moody chart is nothing but a more convenient graphical solution to the same relationship.

    Minor Losses

    Minor losses arise from localized disturbances to the flow caused by components such as:
    • bends,
    • valves,
    • fittings,
    • entrances and exits,
    • contractions and expansions.
    Visualization of minor loss sources
    These components disturb the velocity field and generate additional mixing and energy dissipation. Minor losses are commonly modeled using a loss coefficient \( K \):
    $$ h_L = K\frac{V^2}{2g} $$
    or equivalently,
    $$ \Delta p = K\frac{\rho V^2}{2} $$
    The coefficient \( K \) depends on the geometry of the component and is usually determined experimentally. Engineers can find the values of these loss coefficients in lookup tables or from manufacturer data. Below is a table providing rough guidance.
    Component Typical \( K \) Value
    Sharp-edged entrance 0.5
    Well-rounded entrance 0.04
    Pipe exit 1.0
    Standard 90\( ^\circ \) elbow 0.3 -- 1.5
    45\( ^\circ \) elbow 0.2 -- 0.4
    Fully open gate valve 0.15
    Fully open globe valve 10
    Fully open angle valve 2
    Fully open ball valve 0.05
    Sudden expansion \( \left(1-\dfrac{A_1}{A_2}\right)^2 \)
    Sudden contraction 0.4 -- 1.0
    Typical minor loss coefficients. Actual values vary with geometry and manufacturer specifications.

    Total System Losses

    For practical pipe systems, total head loss is often estimated by summing both major and minor losses:
    $$ h_{L,\text{total}} = h_f+\sum h_L $$
    or
    $$ h_{L,\text{total}} = f\frac{L}{D}\frac{V^2}{2g} + \sum K\frac{V^2}{2g} $$
    These relationships are widely used in pipe network design and pump selection problems.

    Example: Total Losses in a Simple Pipe System

    Consider water flowing steadily through a horizontal pipe of length \( L \) and diameter \( D \). The pipe contains one entrance, two elbows, one valve, and one exit. The average velocity in the pipe is \( V \). The goal is to estimate the total pressure drop caused by both major and minor losses.

    Symbolic Setup

    The major loss due to wall friction is:
    $$ h_f = f\frac{L}{D}\frac{V^2}{2g} $$
    The minor losses are written using loss coefficients:
    $$ h_m = \sum K\frac{V^2}{2g} $$
    For this pipe system:
    $$ \sum K = K_{\text{entrance}} + 2K_{\text{elbow}} + K_{\text{valve}} + K_{\text{exit}} $$
    Thus, the total head loss is:
    $$ h_{L,\text{total}} = \left( f\frac{L}{D} + K_{\text{entrance}} + 2K_{\text{elbow}} + K_{\text{valve}} + K_{\text{exit}} \right) \frac{V^2}{2g} $$
    The corresponding pressure drop is:
    $$ \Delta p = \rho g h_{L,\text{total}} $$
    or:
    $$ \Delta p = \left( f\frac{L}{D} + \sum K \right) \frac{\rho V^2}{2} $$

    Numerical Example

    Suppose:
    $$ L=20\ \text{m}, \qquad D=0.05\ \text{m}, \qquad V=2.0\ \text{m/s} $$
    $$ f=0.024, \qquad \rho=1000\ \text{kg/m}^3 $$
    and the loss coefficients are:
    $$ K_{\text{entrance}}=0.5, \qquad K_{\text{elbow}}=0.3, \qquad K_{\text{valve}}=2.0, \qquad K_{\text{exit}}=1.0 $$
    First compute the total minor-loss coefficient:
    $$ \sum K = 0.5+2(0.3)+2.0+1.0 = 4.1 $$
    The major-loss coefficient is:
    $$ f\frac{L}{D} = 0.024\left(\frac{20}{0.05}\right) = 9.6 $$
    Therefore:
    $$ f\frac{L}{D}+\sum K = 9.6+4.1 = 13.7 $$
    The dynamic pressure term is:
    $$ \frac{\rho V^2}{2} = \frac{1000(2.0)^2}{2} = 2000\ \text{Pa} $$
    Thus, the total pressure drop is:
    $$ \Delta p = 13.7(2000) = 27400\ \text{Pa} $$
    or:
    $$ \Delta p = 27.4\ \text{kPa} $$
    This example shows that the total pressure drop comes from both pipe-wall friction and localized losses from components. Even though they are called minor losses, the fittings and valve contribute a significant portion of the overall loss.