Chapter 1, Review of Mathematical Principles and Applications in Food Processing Video Solutions, Fundamentals of Food Process Engineering

Problem 1

A warehouse having a volume of $10,000 \mathrm{ft}^3$ and a floor area of $1000 \mathrm{ft}^2$ is to be built. The cost of constructing the floor is $$\$ 6.00 / \mathrm{ft}^2$$, the cost of the roof is $$\$ 10.00 / \mathrm{ft}^2$$, and the cost of the walls is $$\$ 20.00 / \mathrm{ft}^2$$. If $\mathrm{W}$ is the width, $\mathrm{L}$ the length, and $\mathrm{H}$ the height of the building, what should the dimensions be such that the cost is minimal?
1.3.

Pawan Yadav

Numerade Educator

01:17

Problem 1

Determine the maximum and minimum value of the following function and prove that the points are maximum or minimum.
$$
C=\frac{315+52.5 T}{(0.21 T-0.76)^{0.5}-1.61}
$$

Jonathon Brumley

Numerade Educator

01:17

Problem 1

Determine the maximum and minimum value of the following function and prove that the points are maximum or minimum.
$$
\mathrm{C}=\frac{315+52.5 \mathrm{~T}}{(0.21 \mathrm{~T}-0.76)^{0.5}-1.61}
$$

Jonathon Brumley

Numerade Educator

07:11

Problem 2

A warehouse having a volume of $10,000 \mathrm{ft}^3$ and a floor area of $$1000 \mathrm{ft}^2$$ is to be built. The cost of constructing the floor is $$\$ 6.00 / \mathrm{ft}^2$$, the cost of the roof is $$\$ 10.00 / \mathrm{t}^2$$, and the cost of the walls is $$\$ 20.00 / \mathrm{ft}^2$$. If $\mathrm{W}$ is the width, $\mathrm{L}$ the length, and $\mathrm{H}$ the height of the building, what should the dimensions be such that the cost is minimal?

Pawan Yadav

Numerade Educator

01:55

Problem 3

Calculate the maximum or minimum value of the following expression. Show that the calculated value is maximum or minimum.
$$
\mathrm{q}=\frac{240 \mathrm{R}}{-0.02+20 \ln (0.5 / \mathrm{R})+0.02 \mathrm{R}}
$$

Daniel Nolan

Numerade Educator

01:22

Problem 4

Calculate the maximum (or minimum value) of $y$.
$$
\begin{aligned}
& y=2 X^2+0.5 R+x+3 \\
& X=0.5 R
\end{aligned}
$$

Ankit Gupta

Numerade Educator

01:24

Problem 5

Calculate the maximum velocity of a fluid flowing inside a pipe expressed in terms of the average velocity.
The point velocity (velocity at any point $r$ measured from the center) is
$$
V=\left[\frac{\Delta p}{2 L K}\right]^{1 / n}\left[1-\left[\frac{r}{R}\right]^{(n+1) / n}\right]\left[\frac{n}{n+1}\right][R]^{(n+1) / n}
$$
The average velocity is
$$
\bar{V}=\left[\frac{\Delta P}{2 L K}\right]^{1 / n}\left[\frac{n}{3 n+1}\right][R]^{(m+1) / n}
$$

Dading Chen

Numerade Educator

02:06

Problem 6

The rate of evaporation of component A divided by the rate of evaporation of component $B$ in a mixture containing $\mathrm{A}$ moles of $\mathrm{A}$ and $\mathrm{B}$ moles of $\mathrm{B}$ is directly proportional to $\mathrm{A} / \mathrm{B}$. If a mixture originally contained 5 moles of $A$ and 3 moles of $B$ and the rate of evaporation of $A$ is 5 moles/h and of $\mathrm{B}$ is $2.6 \mathrm{moles} / \mathrm{h}$, derive an equation for the concentration of A relative to that of $\mathrm{B}$.

Ronald Prasad

Numerade Educator

00:54

Problem 7

The rate of production of ethanol in a fermentor is directly proportional to the number of cells of yeast present. At a cell mass of $14 \mathrm{~g}$ yeast/liter, ethanol production rate is $18 \mathrm{~g}$ ethanol/ $(\mathrm{L} \cong \mathrm{h}$ ). If a batch fermentor is inoculated with $0.1 \mathrm{~g}$ of yeast $/ \mathrm{L}$, and the generation time of the yeast is 1.5 hours, what will be the ethanol production after 10 hours of fermentation. Assume no alcohol-induced inhibition of yeast growth and that yeast doubles in cell mass with each generation time.

Nicole Smina

Numerade Educator

05:00

Problem 8

The work required to compress a gas is given by the following integral:
$$
\mathrm{W}=\int_{\mathrm{V}_1}^{\mathrm{V}_1} \mathrm{PdV}
$$
Van der Waals equation of state for a gas is is follows:
$$
\mathrm{P}=\frac{\mathrm{nRT}}{\mathrm{V}-\mathrm{nb}}-\frac{\mathrm{n}^2 \mathrm{a}}{\mathrm{V}^2}
$$
$\mathrm{R}=82.06\left(\mathrm{~cm}^3 \cong \mathrm{atm}\right) /(\mathrm{g}$ mole $\cong \mathrm{K}), \mathrm{b}=36.6 \mathrm{~cm}^3 / \mathrm{g}$ mole, and $\mathrm{a}=1.33 \times 10^6$ (atm $\cong$ $\left.\mathrm{cm}^6\right) /(\mathrm{g}$ mole $) 2$, when $\mathrm{P}$ is in atm, $\mathrm{T}$ in Kelvin, and $\mathrm{V}$ in $\mathrm{cm}^3$. Calculate the work done when the pressure of the gas is increased from 1 to $10 \mathrm{~atm}$. There was originally I liter of gas at $293 \mathrm{~K}$.

Rowan Ahmed

Numerade Educator

01:32

Problem 9

Write a computer program in BASIC which can be used to calculate a value for the water activity of a solution containing two sugars having mass fractions (percentage composition by weight expressed as a decimal) $x_1$ and $x_2$. $x_1$ is sucrose (molecular weight $342 ; k=2.7$ ); $x_2$ is glucose (molecular weight $180, \mathrm{k}=0.5$ ). The water activity of the mixture is
$$
\mathrm{a}_w=\left(\mathrm{a}_{\mathrm{w} 1}\right)^0\left(\mathrm{a}_{w 2}\right)^0
$$
where $\left(\mathrm{a}_{\mathrm{w} 1}\right)^0$ is water activity of a solution containing all the water in the mixture and component 1 ; and $\left(a_{w_2}\right)^0$ is water activity of a solution containing all the water in the mixture and component 2.
$$
\log _{10}\left[\frac{\left(a_{e i}\right)^0}{f_i}\right]=-k_i\left(1-f_i\right)^2
$$
where $f_i$ is mole fraction of component $i$ in a solution containing only component $i$ and all the water present in the mixture.
$$
f_i=1-\left[\frac{x_i / M_i}{\left(x_i / M_i\right)+\left(1-x_1-x_2\right) / 18}\right]
$$

Prem Bijarniya

Numerade Educator

00:32

Problem 10

Determine the slope of the following functions:
$$
\begin{array}{cl}
f(x)=2 x^4-3 x^2+7=0 & \text { at } x=2 \\
x y=(0.5 x+3)(x+2) & \text { at } x=1
\end{array}
$$

Mike Gaerlan

Numerade Educator

01:25

Problem 11

Determine the maximum and minimum value of the functions in Problem 10 above.

Erika Bustos

Numerade Educator

01:25

Problem 12

Determine the slope of the following function at the indicated point.
$$
x=(0.5 x y+3)\left(3+x^2\right) \quad \text { at } x=1
$$

Gregory Higby

Numerade Educator

04:31

Problem 13

Construct a spreadsheet that can be used to determine the boiling temperature of a liquid in an evaporator as it is being concentrated. The boiling point of a liquid is the temperature at which the vapor pressure equals the atmospheric pressure. A solution will exhibit a boiling point rise because the solute will lower the water activity resulting in a higher temperature to be reached before boiling occurs. The vapor pressure of water $\left(\mathrm{P}^0\right)$ as a function of temperature is expressed by the following equation:
$$
\ln \left(\mathrm{P}^0\right)=\left(-\frac{H}{R}\right)\left(\frac{1}{T}\right)+C
$$
where $\mathrm{P}^0$ is the vapor pressure in kilopascals, $\mathrm{H} / \mathrm{R}$ is the ratio of the latent heat of vaporization and the gas constant, $\mathrm{C}$ is a constant, and $\mathrm{T}$ is the absolute temperature in degrees Kelvin. The value for $H / R$ is 4950 and $C$ is 17.86 .
Atmospheric pressure is 101 kilopascals. The vapor pressure of a solution is $\mathrm{P}=\mathrm{a}_{\mathrm{w}} \mathrm{P}^0$. The water activity $\left(\mathrm{a}_{\mathrm{w}}\right)$ is given by:
$$
\log _{10}\left(\frac{a_w}{f_1}\right)=-k\left(1-f_1\right)^2
$$
$f_1$ is the mole fraction of water and is calculated by:
$$
f_1=\frac{\left(1-x_1\right) / 18}{\left(1-x_1\right) / 18+x_1 / M_1}
$$
$x_1$ is the mass fraction of solute. Assume the solute is only sucrose with a molecular weight $\left(\mathrm{M}_1\right)$ of 342 and a $\mathrm{k}$ value of 2.7 . Have the program display the value of the boiling point when the sucrose concentration is $20 \%$ and at increasing concentrations in $5 \%$ intervals to a final concentration of $60 \%$.

Lottie Adams

Numerade Educator

01:20

Problem 14

Calculate a value of $x$ that would give the minimum value for $S$ in the following expression:
$$
\mathrm{S}=50\left(9.522 \times 10^{-6}\right)-\frac{50}{(50 \mathrm{X}+1)}\left(9.522 \times 10^{-6}\right)-\frac{60(0.22) \mathrm{X}}{30(24)}
$$
Show that the function is a minimum or maximum.

Vishnu Rathnam

Numerade Educator

03:03

Problem 15

In the dehydration of diced potatoes, the following weights were recorded at various times in the process when dehydration rate was the slowest.
$\mathrm{t}=6$ hours from start of drying: weight $=2350 \mathrm{~g}$
$\mathrm{t}=8$ hours from start of drying: weight is $2275 \mathrm{~g}$; moisture content $=15.6 \%$.
In this range of moisture content, the drying rate is proportional to the moisture content, expressed in mathematical form as follows:
$$
-\frac{\mathrm{dW}}{\mathrm{dt}}=\mathrm{kW}
$$
$\mathrm{W}$ is the moisture content in $\mathrm{g}$ water/g dry matter, and $\mathrm{k}$ is a constant.
(a) Derive an equation for the moisture content, W, as a function of time, $t$, which satisfies the experimental conditions given above.
(b) How long would it take for a product to be dehydrated from $22.5 \%$ to $12.5 \%$ moisture?
$$
\frac{\mathrm{N}_0}{\mathrm{~N}}=\frac{\mathrm{R}^2 \overline{\mathrm{V}}}{2}\left[\frac{1}{\int_0^{\mathrm{R}} \mathrm{rV}_{\mathrm{r}}[10]^{-\mathrm{L} /(\mathrm{V}, \mathrm{D}) \mathrm{dr}}}\right]
$$

Rashmi Sinha

Numerade Educator

15:18

Problem 16

Write a computer program in BASIC that will solve the following integral: where
$$
V_t=\bar{V}\left[\frac{3 n+1}{n+1}\right]\left[1-\frac{r}{R}^{(n+1) / n}\right]
$$
Given: $\nabla=2, R=0.02, n=0.6, L=10, D=1$. Use the trapezoidal rule in evaluating the integral and use increments in $r$ of 0.0001 .

Oswaldo Jiménez

Numerade Educator

01:45

Problem 17

Linearize the following equations. In each case, indicate which function should be plotted as the independent and dependent variable to obtain a linear plot. What is the slope and intercept of each plot?
(a) $\log \left(\mathrm{a}_{\mathrm{w}} / \mathrm{X}_{\mathrm{i}}\right)=-\mathrm{k}\left(1-\mathrm{X}_{\mathrm{i}}\right)^2$ where $\mathrm{a}_{\mathrm{w}}$ and $\mathrm{X}_{\mathrm{i}}$ are variables.
(b) $(1 / \mathrm{x})=(\mathrm{a}+\mathrm{b}) / 2 \mathrm{y}+\mathrm{c} / 4 \mathrm{y}$ where $\mathrm{x}$ and $\mathrm{y}$ are variables and $\mathrm{a}, \mathrm{b}$, and $\mathrm{c}$ are constants.
(c) $\mathrm{N}=\mathrm{N}_0(10)^{-(t / D)}$ where $\mathrm{N}$ and $\mathrm{t}$ are variables and $\mathrm{N}_0$ and $\mathrm{D}$ are constants.

Prajwal Holikatti

Numerade Educator

01:56

Problem 18

The velocity of an enzyme-catalyzed reaction (V) expressed as a function of the substrate
$$
\mathrm{V}=\frac{\mathrm{V}_{\max }(\mathrm{S})}{\mathrm{K}_{\mathrm{m}}+\mathrm{S}}
$$
concentration (S) is given by the following equation:
where $V_{\max }$ is the maximum reaction rate and $\mathrm{K}_{\mathrm{m}}$ is a constant for the reaction. Linearize the equation and determine the independent and dependent variables to be plotted to obtain $\mathrm{V}_{\text {mas }}$ and $\mathrm{K}_{\mathrm{m}}$ from the slope and intercept.

Cinsy Krehbiel

Numerade Educator

04:32

Problem 19

The temperature $(\mathrm{T})$ of a refrigerant during compression in a refrigeration system increases with pressure $(\mathrm{P})$ as shown in the following equation. $\mathrm{P}_1$ and $\mathrm{T}_1$ are reference temperature and pressure and are considered constants.
$$
\frac{\mathrm{T}}{\mathrm{T}_1}=\left[\frac{\mathrm{P}}{\mathrm{P}_1}\right]^{(\mathrm{k}-\mathrm{1}) / \mathrm{k}}
$$
The following data are available on the temperature and pressure of a refrigerant during compression:
$\begin{array}{crrrrr}\mathrm{T}(\mathrm{ER}) & 450 & 510 & 550 & 608 & 640 \\ \mathrm{P}\left(\mathrm{lb}_{\mathrm{f}} / \mathrm{in}^2\right) & 4 & 6 & 10 & 20 & 30\end{array}$
Linearize the above equation, perform a linear regression, and determine the constant $k$ for this refrigerant. Calculate the value of $\mathrm{k}$ by nonlinear curve fitting.

Km Neeraj

Numerade Educator

06:58

Problem 20

The heat of respiration of leafy greens as a function of time when the temperature is changing during a cooling operation is as follows:
$$
q=\int_0^t 0.009854[e]^{0.073 T} d t
$$
where $\mathrm{q}$ is in $\mathrm{BTU} /(\mathrm{b}), \mathrm{T}$ is in ${ }^{\circ} \mathrm{F}$, and $\mathrm{t}$ is time in hours. The temperature of a box of spinach containing $50 \mathrm{lb}$, originally at $110^{\circ} \mathrm{F}$, will cool down exponentially when placed in a refrigerated room at $35^{\circ} \mathrm{F}$ according to:
$$
\mathrm{T}=35+(75) \exp (-\mathrm{t} / 5)
$$
where $\mathrm{t}$ is the time in hours. Evaluate this integral using a spreadsheet and determine the total heat generated by the spinach as the box cools down from $110^{\circ} \mathrm{F}$ to $40^{\circ} \mathrm{F}$. Use Simpson's rule to determine the value of the integral.

Jamie Fife

Numerade Educator

02:07

Problem 21

An immobilized enzyme reactor must have the enzyme regenerated periodically because of the decay in the activity of the enzyme. The enzyme will convert $87 \%$ of the substrate to product within the first day of operation, and this conversion changes to $0.87 /(t)^{0.82}$ after the first day. If the feed rate is $150 \mathrm{lb}$ of substrate per day, the amount of product formed after $\mathrm{t}$ days of operation

Narayan Hari

Numerade Educator