Solve and plot the graphs for various business case study

Assignment #1

Answer all questions. Although many calculations can be done by hand, I strongly suggest you use Microsoft Excel spreadsheet for problem solving and plotting the graphs. The assignment must be type written in Microsoft Word with proper justifications of your answers (provide sample calculations, show formulas, explain critical steps, etc.). These details are important for the TAs; if they can’t follow your step/rationale, they will not be able to give you the points when grading the assignment.

Question 1

Water barrier properties of food packages are often evaluated gravimetrically according to American Society for Testing and Materials’ protocol, such as ASTM D7709. A typical approach involves placing a desiccant inside the test package and sealed. The desiccant would absorb moisture from the package headspace, thereby creating an 0% RH interior environment. During the test, the package is exposed to an atmosphere kept at constant temperature and relative humidity. This creates a concentration gradient that causes the migration of water from the surrounding air into the package headspace. As the desiccant continues to absorb the ingress water, an increase in weight of the package is observed, the rate of which can be used for determining the water vapor transmission rate (WVTR) of the package.

Below are weight data (g) of thermoformed PET cups, sealed with two experimental lidding films A and B. To maintain an interior 0% RH, enough desiccant was placed in the thermoformed packages before sealing them with the test lidding films. The sealed packages were stored at 20^oC and 75% RH condition. Each of these two films was tested five times.

1. Create scatter plots of weight versus time for the two films, respectively (i.e., two graphs; 5 plots each). Label the axes clearly. (10%)

2. Calculate the average and the standard deviation values of WVTR, in g/h.package for lidding A and B. Explain the possible causes of the differences in the mean and standard deviation values for the two lidding. Which lidding would you use for a moisture-sensitive product? Justify your answer. (20%)

3. For the lidding that you selected in part (ii), if the package containing a product is maintained at 50% RH and the exterior environment at 75% RH, calculate the amount of moisture that would permeate through the sealed package in one year. (10%)

Question 2

Consider an aseptic oat milk beverage (950 mL) packaged in the TetraBrik carton with a dimension of L x W x H. The unfolded paperboard blank of the carton with scores lines is shown below. The shaded regions represent seal flaps required for forming the carton. We are interested to determine the dimensions of the carton that provide a minimal packaging material usage.

1. If W and S are 5.8 cm and 0.8 cm, respectively, determine the dimensions of L and H that would result in the smallest area, assuming that the package has a headspace volume of 50 mL. Hint: write an equation area of the unfolded carton as a function of L; create a scatter (X-Y) plot of area versus L; determine the L value when the area is the smallest. (30%)

2. From a teardown analysis of an actual carton package of a commercial product, the L x W x H dimensions are 8.9 cm, 5.8 cm, and 19.4 cm, respectively. Calculate the surface area (remember to include the seal flaps) based on these dimensions and comment on whether the commercial package is optimized in terms of paperboard utilization. Justify your answer. (10 %).

3. 

Question 3

The graph on the right below shows tensile test data for nylon and LDPE film specimens (40

m

m thickness; 2 cm width; 5 cm initial length) stretched to the break point. The graph on the left has the same data with the initial deformation range magnified to show the graphical details. For each of these polymers, the films were cut from the same web in either machine or transverse directions (labelled as 1 or 2).

1. Using the graphs provided, calculate the ultimate tensile strength (in MPa), elongation at break (in %), apparent modulus (in MPa) and toughness (in J/m³) values for all the four films. Hint: The toughness, which is equivalent to the area under the stress-strain curve, can be estimated from number of squares between the curve and X-axis or using basic geometries. (30%)

2. Which film, nylon or LDPE, is likely to be uniaxial oriented? Justify your answers. (5%)

3. Between nylon and LDPE, which film is tougher, and which is stiffer? Justify your answers. (5%)