ENCE 489G/621, Homework 2

1. Use the following figure to derive the capillary rise equation of non-ideal contact between water and a glass tube surface. In the figure, r is the radius of the tube, R is the  curvature of the air-water interface,γ is the contact angle.
2. Take a napkin from your kitchen/diner and roll it up.  Mark on the napkin the initial level of the water that you intend to use. Place the napkin in a glass and add water to the level that you marked earlier. Let the napkin stay in the water for 10 minutes. Measure the rise of water in the napkin. Assume a contact angle of 15^o and calculate the equivalent pore radius of the napkin. Repeat the same experiment for a piece of notebook paper. What is your conclusion about the texture of the two types of paper? Is the 15^o contact angle a reasonable assumption for both types of paper? Why?
   
   
3. A collection of capillary tubes is randomly inserted into a pipe with a cross section of area 25 cm².  The lengths of the tubes and the pipe are uniformly 200 cm. The frequency of tubes and their corresponding radii are listed in the following table:
   

   radius (cm)	frequency
   0.0008185	1
   0.0013559	4
   0.0021006	16
   0.0039242	44
   0.0059515	91
   0.0093489	149
   0.0175649	191
   0.0248351	191
   0.0450413	149
   0.0736017	91
   0.1346353	44
   0.198646	16
   0.312416	4
   0.613039	1

   Use a water surface tension of 72.2 erg/cm² (cm•g•cm/s²•cm^-2) and do the following calculations.

   • Calculate the porosity of the bundle of tubes. Assume that the space outside the tubes are filled with non-porous solid grains.
     
   • Assuming that the contact between the tube and water is ideal (contact angle = 0), calculate the capillary rise for each tube radius class.
   • Calculate the water content and degree of saturation. Designate this water content as the water content corresponding to the largest tube radius and the lowest capillary rise.
     
   • Assume that one may remove only the water in the tubes with the largest radius by a vacuum but not the water in the rest of the tube classes. Recalculate the water content and degree of saturation for the bundle of tubes. Designate this water content as the water content corresponding to the second largest tube radius and the second lowest capillary rise.
   • Repeat the above calculation until the tubes with the smallest radius. Plot the capillary rise as a function of the water content in an x-y plot (x is water content and y is capillary rise), one with both axes in linear scale and the other with the capillary rise in logarithmic scale.
   • What is the residual water content in the figure above?