Question

Problem 1. A lemonade recipe calls for two basic steps: creating a simple syrup from 1.75 cups white sugar and 1.00 cup of boiling water, then cooling this and mixing it with 1.50 cups lemon juice and 7.00 cups of cold water.

(a) Draw a flowchart for this batch process and conduct a degree of freedom analysis to confirm that we have enough information to determine the outputs from this process.

(b) Rewrite this recipe so that all input quantities are expressed as masses in grams. Determine the mass and volume of lemonade produced. Assume that the apparent (bulk) density of white sugar is 0.95 g/cm3 , the densities of cold and boiling water are basically the same 1.00 g/mL, and the density of lemon juice is 1.03 g/mL. Assume the volumes of water and lemon juice are additive, but that the sugar does not change the volume of the mixture.

(c) Say I want to mass-produce lemonade on a modest scale and can process 500 lbm of sugar and up to 1000 lemons per day. If a lemon contains 3 tablespoons of juice, what is my limiting recipe component? How much water would I need? How much lemonade can I make per day, in gallons? There are 16 tablespoons in a cup, and 16 cups in a gallon.

Answer 1

The flowsheet of the process is shown

Number of unknowns =2 ( mass of lemonade ) and (volume of lemonade)

DOF analysis

Sugar does not change volume of mixture

Volume of lemonade = volume of hot water + volume of lemon juice + volume of cold water

Mass of lemonade = mass of white sugar + mass of hot water + mass of cold water+ mass of lemon juice

Number of equations = 2

DOF = 2-2 = 0

Hence enough information is there to solve the problem

Given data

16 cups = 1 gallon

From Perry handbook

1 gallon = 3.785 L

4.2272 cups = 1 L

1000 cm3 = 1 L

B) Volume of white sugar = 1.75 cups

= 1.75/4.2272 = 0.413985 L = 413.985 cm3

Density of white sugar = 0.95 g/cm3

Mass of white sugar = 0.95(413.985) = 393.286 g

Volume of boiling water = 1 cup

= 1/4.2272 = 0.23656 L = 236.563 cm3

Density of water = 1 g/cm3

Mass of boiling water = 236.563 g

Volume of lemon juice = 1.50 cups

= 1.50/4.2272 = 0.35484 L = 354.844 cm3

Density of lemon juice = 1.03 g/mL

Mass of lemon juice = 1.03(354.844) = 365.490 g

Volume of cold water = 7 cup

= 7/4.2272 = 1.6559 L = 1655.942 cm3

Density of water = 1 g/cm3

Mass of cold water = 1(1655.942) = 1655.942 g

Sugar does not change volume of the mixture

Volume of lemon juice and water are additive

Volume of lemonade =

volume of boiling water + volume of lemon juice + volume of cold water =

0.23656+0.35484+1.6559 = 2.2473 Litres

Volume of lemonade produced = 2.2473 litres = 0.5937 gallons

Mass of lemonade = mass of sugar + mass of boiling water + mass of lemon juice + mass of cold water = 393.286+236.563+365.490+1655.942 =

2651.281 g = 2.6512 Kg

C)

Sugar produced = 500 lbm

Mass of sugar in kg = 500(0.454) =227 Kg

Lemons made = 1000 lemons

There are 3 table spoons of juice in a lemon

Total table spoons of juice = 1000(3) = 3000 table spoons

16 table spoons = 1 cup

Cups of lemon juice = (3000/16) = 187.5 cups

16 cups = 1 gallon

Volume of lemon juice = 187.5/16 =

11.71875 gallons = 44.3554 L

Density of lemon juice = 1.03 g/cm3

Mass of lemon juice = 1.03(1000) (44.3554) = 45686.062 g

From part A

Mass of white sugar : Mass of lemon juice

= 393.286 : 365.490 = 1.0760

In this case

Mass of white sugar : mass of lemon juice =

227000: 45686.062 = 4.9686

Hence the limiting recipe is lemon juice

Limiting component = lemon juice

Volume of lemon juice =44.3554 L

From part B

Volume fraction of lemon juice in lemonade =

(0.35484/2.2473) = 0.15789

Total volume of lemonade produced =

(44.3554/0.15789 ) = 280.925 Litres

Volume of lemonade = 74.220 gallons

From part B

Mass fraction of cold and hot water in lemonade = (0.23656+1.6559) /2.2473 = 0.84210

Volume of lemonade produced =

74.220 gallons /day

Volume of water needed = 74.220(0.84210) = 62.50 gallons/day

Please upvote if helpful