Example 10.5-2 Flow of Nonisothermal Films

First, we will look at Figs. 2.2-1 and 2. Fig. 2.2-1 illustrates the end effects of the falling film.

Fig. 2.2-2 illustrates the flow of the viscous liquid film under gravitational influence with no rippling. The energy balance is made over a slice of thickness delta(x). In this diagram, the y-axis points outward from the screen.

We will start by going back to chapter 2 in BS&L to look at the derivation of the equations that govern flow in a film on an inclined plane surface.

 The properties are expressed in cgs units for water over the range of temperatures: 0 to 100C. The viscosity was determined at the average temoerature: 50C. Here are the velocities at various points in the film in cm/s.

Here is the first case where variable viscosity is treated. Its dependence on position in the film is assumed to be exponential as specified by eq. 2.2-22. The stress can be found just as for the first case, but the equation that determines the velocity is more complex. 

 Let's see if that function agrees with equation 2.2-24.

 It does check and it has the right limit as the parameter alpha goes to 0. Now we can go on to Example 10.5-2. The temperature is a simple linear function of distance in the film. Some problems were encountered in trying to define a function of a function to get the dependence of mu on distance so a procedure was used to do this.

 The approximation used for mu as a function of distance was needed so the equation for velocity in a film with varying viscosity from section 2.2 in Example 2.2-2 could be used. We compared the approximations for mu for water over the temperature range 0 to 100C. The function muapp gave much larger errors in viscosity than the mu function.

 
Position: x/delta
0.5
0.6
Temperature C
50
60
mu from Table 1.1-1
~0.566
0.4665
mu from eq 10.5-19
0.7145
0.5949
mu from eq. 10.5-19a
0.6201
0.5213

 

 

 Those velocities were found using the better mu approximation: eq. 10.5-19a. Now we will compare it with the approximation 10.5-19c (or 10.5-19.)

 

 

 Certainly both approximations are much better than the use of the average temperature to determine mu and then use the isothermal assumption.

x/delta
0.0
0.5
0.8
0.9
1.0

Isothermal using mu at average temperature

24.50
18.38
8.82
4.66
0.00

Nonisothermal using approximation in 10.5-19

28.69
25.03
14.73
8.44
0.00

Nonisothermal using fit of data with eq. 10.5-19a

31.05
26.89
15.33
8.13
0.00