2.9. SHELL MOMENTUM BALANCE AND VELOCITY PROFILE IN LAMINAR FLOW

2.9A. Introduction

In Section 2.8 we analyzed momentum balances using an overall, macroscopic control volume. From this we obtained the total or overall changes in momentum crossing the control surface. This overall momentum balance did not tell us the details of what happens inside the control volume. In the present section we analyze a small control volume and then shrink this control volume to differential size. In doing this we make a shell momentum balance using the momentum-balance concepts of the preceding section, and then, using the equation for the definition of viscosity, we obtain an expression for the velocity profile inside the enclosure and the pressure drop. The equations are derived for flow systems of simple geometry in laminar flow at steady state.

In many engineering problems a knowledge of the complete velocity profile is not needed, but a knowledge of the maximum velocity, the average velocity, or the shear stress on a surface is needed. In this section we show how to obtain these quantities from the velocity profiles.

2.9B. Shell Momentum Balance Inside a Pipe

Engineers often deal with the flow of fluids inside a circular conduit or pipe. In Fig. 2.9-1 we have a horizontal section of pipe in which an incompressible Newtonian fluid is flowing in one-dimensional, steady-state, laminar flow. The flow is fully developed; that is, it is not influenced by entrance effects and the velocity profile does not vary along the axis of flow in the x direction.

The cylindrical control volume is a shell with an inside radius r, thickness Δr, and length Δx. At steady state the conservation of momentum, Eq. (2.8-3), becomes as follows: sum of forces acting on control volume = rate of momentum out − rate of momentum into volume. The pressure forces become, from Eq. (2.8-17),

Equation 2.9-1

The shear force or drag force acting on the cylindrical surface at the radius r is the shear stress τ_rx times the area 2πr Δx. However, this can also be considered as the rate of momentum flow into the cylindrical surface of the shell as described by Eq. (2.4-9). Hence, the net rate of momentum efflux is the rate of momentum out − rate of momentum in and is

Equation 2.9-2

The net convective momentum flux across the annular surface at x and x + Δx is zero, since the flow is fully developed and the terms are independent of x. This is true since ν_x at x is equal to ν_x at x + Δx.

Equating Eq. (2.9-1) to (2.9-2) and rearranging,

Equation 2.9-3

In fully developed flow, the pressure gradient (Δp/Δx) is constant and becomes (Δp/L), where Δp is the pressure drop for a pipe of length L. Letting Δr approach zero, we obtain

Equation 2.9-4

Separating variables and integrating,

Equation 2.9-5

The constant of integration C₁ must be zero if the momentum flux is not infinite at r = 0. Hence,

Equation 2.9-6

This means that the momentum flux varies linearly with the radius, as shown in Fig. 2.9-2, and the maximum value occurs at r = R at the wall.

Substituting Newton's law of viscosity,

Equation 2.9-7

into Eq. (2.9-6), we obtain the following differential equation for the velocity:

Equation 2.9-8

Integrating using the boundary condition that at the wall, ν_x = 0 at r = R, we obtain the equation for the velocity distribution:

Equation 2.9-9

This result shows us that the velocity distribution is parabolic, as shown in Fig. 2.9-2.

The average velocity ν_xav for a cross section is found by summing up all the velocities over the cross section and dividing by the cross-sectional area, as in Eq. (2.6-17). Following the procedure given in Example 2.6-3, where dA = rdrdθ and A = πR²,

Equation 2.9-10

Combining Eqs. (2.9-9) and (2.9-10) and integrating,

Equation 2.9-11

where diameter D = 2R. Hence, Eq. (2.9-11), which is the Hagen–Poiseuille equation, relates the pressure drop and average velocity for laminar flow in a horizontal pipe.

The maximum velocity for a pipe is found from Eq. (2.9-9) and occurs at r = 0:

Equation 2.9-12

Combining Eqs. (2.9-11) and (2.9-12), we find that

Equation 2.9-13

Also, dividing Eq. (2.9-9) by (2.9-11),

Equation 2.9-14

2.9C. Shell Momentum Balance for Falling Film

We now use an approach similar to that used for laminar flow inside a pipe for the case of flow of a fluid as a film in laminar flow down a vertical surface. Falling films have been used to study various phenomena in mass transfer, coatings on surfaces, and so on. The control volume for the falling film is shown in Fig. 2.9-3a, where the shell of fluid considered is Δx thick and has a length of L in the vertical z direction. This region is sufficiently far from the entrance and exit regions so that the flow is not affected by these regions. This means the velocity ν_z(x) does not depend on position z.

To start we set up a momentum balance in the z direction over a system Δx thick, bounded in the z direction by the planes z = 0 and z = L, and extending a distance W in the y direction. First, we consider the momentum flux due to molecular transport. The rate of momentum out − rate of momentum in is the momentum flux at point x + Δx minus that at x times the area LW:

Equation 2.9-15

The net convective momentum flux is the rate of momentum entering the area ΔxW at z = L minus that leaving at z = 0. This net efflux is equal to 0, since ν_z at z = 0 is equal to ν_z at z = L for each value of x:

Equation 2.9-16

The gravity force acting on the fluid is

Equation 2.9-17

Then, using Eq. (2.8-3) for the conservation of momentum at steady state,

Equation 2.9-18

Rearranging Eq. (2.9-18) and letting Δx → 0,

Equation 2.9-19

Equation 2.9-20

Integrating using the boundary conditions at x = 0, τ_xz = 0 at the free liquid surface and at x = x, τ_xz = τ_xz,

Equation 2.9-21

This means the momentum-flux profile is linear, as shown in Fig. 2.9-3b, and the maximum value is at the wall. For a Newtonian fluid using Newton's law of viscosity,

Equation 2.9-22

Combining Eqs. (2.9-21) and (2.9-22) we obtain the following differential equation for the velocity:

Equation 2.9-23

Separating variables and integrating gives

Equation 2.9-24

Using the boundary condition that ν_z = 0 at x = δ, C₁ = (ρg/2μ)δ². Hence, the velocity-distribution equation becomes

Equation 2.9-25

This means the velocity profile is parabolic, as shown in Fig. 2.9-3b. The maximum velocity occurs at x = 0 in Eq. (2.9-25) and is

Equation 2.9-26

The average velocity can be found by using Eq. (2.6-17):

Equation 2.9-27

Substituting Eq. (2.9-25) into (2.9-27) and integrating,

Equation 2.9-28

Combining Eqs. (2.9-26) and (2.9-28), we obtain . The volumetric flow rate q is obtained by multiplying the average velocity ν_zav times the cross-sectional area δW:

Equation 2.9-29

Often in falling films, the mass rate of flow per unit width of wall Г in kg/s · m is defined as Г = ρδν_zav and a Reynolds number is defined as

Equation 2.9-30

Laminar flow occurs for N_Re < 1200. Laminar flow with rippling present occurs above a N_Re of 25.

EXAMPLE 2.9-1. Falling Film Velocity and Thickness An oil is flowing down a vertical wall as a film 1.7 mm thick. The oil density is 820 kg/m3 and the viscosity is 0.20 Pa · s. Calculate the mass flow rate per unit width of wall, Г, needed and the Reynolds number. Also calculate the average velocity. Solution: The film thickness is δ = 0.0017 m. Substituting Eq. (2.9-28) into the definition of Г, Equation 2.9-31 Using Eq. (2.9-30), Hence, the film is in laminar flow. Using Eq. (2.9-28),