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Fundamentals of Food Process Engineering

Romeo T. Toledo

Chapter 6

Flow of Fluids - all with Video Answers

Educators

Chapter Questions

02:38

Problem 1

The following data were obtained when tomato catsup was passed through a tube having an inside diameter of $1.384 \mathrm{~cm}$ and a length of $1.22 \mathrm{~m}$.
$$
\begin{array}{cc}
\hline \text { Flow rate }\left(\mathrm{cm}^3 / \mathrm{s}\right) & \left.P \text { (dynes } / \mathrm{cm}^2\right) \\
\hline 107.5 & 50.99 \times 10^4 \\
67.83 & 42.03 \times 10^4 \\
50.89 & 33.07 \times 10^4 \\
40.31 & 29.62 \times 10^4 \\
10.10 & 15.56 \times 10^4 \\
8.80 & 14.49 \times 10^4 \\
33.77 & 31.00 \times 10^4 \\
53.36 & 35.14 \times 10^4 \\
104.41 & 46.85 \times 10^4 \\
\hline
\end{array}
$$
Determine the fluid consistency index $\mathrm{K}$ and the flow behavior index $\mathrm{n}$ of this fluid.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
06:59

Problem 2

Figure 6.30 shows a water tower and the piping system for a small manufacturing plant. If water flows through the system at $40 \mathrm{~L} / \mathrm{min}$, what would be the pressure at point B? The pipes are wrought iron pipes. The water has a density of $998 \mathrm{~kg} / \mathrm{m}^3$ and a viscosity of 0.8 centipoise. The pipe is 1.5 -in. (nominal) wrought iron pipe.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:01

Problem 3

The catsup in Problem $1\left(\mathrm{n}=0.45, \mathrm{~K}=6.61 \mathrm{~Pa} \cdot \mathrm{s}^{\mathrm{n}}\right)$ is to be heated in a shell and tube heat exchanger. The exchanger has a total of 20 tubes $7-\mathrm{m}$ long arranged parallel inside a shell. Each tube is a 3/4-in. outside diameter, 18 gauge heat exchanger tube, (From a table of thickness of sheet metal and tubes, an 18-gauge wall is 0.049 in.) It is possible to arrange the fluid flow pattern by the appropriate selection of heads for the shell, such that number of passes and the number of tubes per pass can be varied. Calculate the pressure drop across the heat exchanger (the pressure at the heat exchanger inlet necessary to push the product through the heat exchanger) if a flow rate of $40 \mathrm{~L}$ of product per $\min$ (density $=1013 \mathrm{~kg} / \mathrm{m}^3$ ) is going through the system for (a) two-pass (10 tubes/pass) and (b) five-pass system (four tubes per pass). Consider only the tube resistance and neglect the resistance at the heat exchanger heads. (FIGURE CAN'T COPY)

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:51

Problem 4

Figure 6.31 shows a de-aerator operated at $381 \mathrm{~mm} \mathrm{Hg}$ vacuum. (Atmospheric pressure is $762 \mathrm{~mm} \mathrm{Hg}$.) It is desired to allow a positive suction flow into the pump. If the fluid has a density of $1040 \mathrm{~kg} / \mathrm{m}^3$, the flow rate is $40 \mathrm{~L} / \mathrm{min}$, and the pipe is 1.5 -in. sanitary pipe, calculate the height " $h$ " the bottom of the de-aerator must be set above the pump level in order that the pressure at the pump intake is at least $5 \mathrm{kPa}$ above atmospheric pressure. The fluid is Newtonian and has a viscosity of 100 centipoises.
[Note: Total length of straight pipe from first elbow below the pump to the entrance to the pump $=4 \mathrm{~m}$. Distance from pump level to horizontal pipe $=0.5 \mathrm{~m}$.]

Nick Johnson
Nick Johnson
Numerade Educator
01:12

Problem 5

Calculate the total equivalent length of 1 -in. wrought iron pipe that would produce a pressure drop of $70 \mathrm{kPa}$ due to fluid friction, for a fluid flowing at the rate of $50 \mathrm{~L} / \mathrm{min}$. The fluid has a density of $988 \mathrm{~kg} / \mathrm{m}^3$ and a viscosity of two centipoises.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:55

Problem 6

Calculate the horsepower required to pump a fluid having a density of $1040 \mathrm{~kg} / \mathrm{m}^3$ at the rate of $40 \mathrm{~L} / \mathrm{min}$ through the system shown in Fig. $6.32 . \Delta \mathrm{P}_{\mathrm{f}} / \rho$ calculated for the system is 120 $\mathrm{J} / \mathrm{kg}$. Atmospheric pressure is $101 \mathrm{kPa}$. (FIGURE CAN'T COPY)

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
02:25

Problem 7

Calculate the average and maximum velocities of a fluid flowing at the rate of $20 \mathrm{~L} / \mathrm{min}$ through a 1.5 -in. sanitary pipe. The fluid has a density of $2030 \mathrm{~kg} / \mathrm{m}^3$ and a viscosity of 50 centipoises. Is the flow laminar or turbulent?

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
01:56

Problem 8

Determine the inside diameter of a tube that could be used in a high-temperature, short-time heater-sterilizer such that orange juice with a viscosity of 3.75 centipoises and a density of $1005 \mathrm{~kg} / \mathrm{m}^3$ would flow at a volumetric flow rate of $4 \mathrm{~L} / \mathrm{min}$ and have a Reynolds number of 2000 while going through the tube.

James Kiss
James Kiss
Numerade Educator
04:34

Problem 9

Calculate the pressure generated at the discharge of a pump that delivers a pudding mix $\left(\rho=995 \mathrm{~kg} / \mathrm{m}^3 ; \mathrm{K}=1.0 \mathrm{~Pa} \cdot \mathrm{s}^{\mathrm{n}}, \mathrm{n}=0.6\right)$ at the rate of $50 \mathrm{~L} / \mathrm{min}$ through $50 \mathrm{~m}$ of a 1.5 -in. straight, level 1.5 -in. stainless steel sanitary pipe. What would be the equivalent viscosity of a Newtonian fluid that would give the same pressure drop? (FIGURE CAN'T COPY)

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:12

Problem 10

What pipe diameter will give a rate of flow of $4 \mathrm{ft} / \mathrm{min}$ for a fluid delivered at the rate of 2 $\mathrm{gal} / \mathrm{min}$ ?

Narayan Hari
Narayan Hari
Numerade Educator
00:59

Problem 11

Calculate the viscosity of a fluid that would allow a pressure drop of $35 \mathrm{kPa}$ over a 5 -m length of $3 / 4$-in. stainless steel sanitary pipe if the fluid is flowing at $2 \mathrm{~L} / \mathrm{min}$ and has a density of 1010 $\mathrm{kg} / \mathrm{m}^3$. Assume laminar flow.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
07:04

Problem 12

A fluid is evaluated for its viscosity using a Brookfield viscometer. Collected data of rotational speed in rev/min and corresponding apparent viscosity in centipoises, respectively, are 20 , $7230 ; 10,12060 ; 4,25200 ; 2,39500$. Is the fluid Newtonian or non-Newtonian. Calculate the flow behavior index, $\mathrm{n}$, for this fluid.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:38

Problem 13

A fluid having a viscosity of $0.05 \mathrm{lb}_{\text {mas }} / \mathrm{ft}$ ( $\mathrm{s}$ ) requires 30 seconds to drain through a capillary viscometer. If this same viscometer is used to determine the viscosity of another fluid and it takes 20 seconds to drain, calculate the viscosity of this fluid. Assume the fluids have the same densities.

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
03:15

Problem 14

Figure 6.33 shows a storage tank for sugar syrup that has a viscosity of 15.2 centiposes at $25^{\circ} \mathrm{C}$, and a density of $1008 \mathrm{~kg} / \mathrm{m}^3$. Friction loss includes an entrance loss to the drain pipe that equals the kinetic energy gain of the fluid, and resistance to flow through the short section of the drain pipe.
(a) Formulate an energy balance equation for the system.
(b) Formulate the continuity equation that represents the increment change in fluid level in the tank as a function of the fluid velocity through the drain pipe.
(c) Solve simultaneously equations formulated in (a) and (b) to calculate the time to drain the tank to a residual level $1 \mathrm{~m}$ from the bottom of the tank.
(d) Calculate the amount of residual fluid in the tank including the film adhering to the side of the tank. (FIGURE CAN'T COPY)

Surjit Tewari
Surjit Tewari
Numerade Educator
05:39

Problem 15

A fluid tested on a tube viscometer $0.75 \mathrm{~cm}$ in diameter and $30 \mathrm{~cm}$ long exhibited a pressure drop of $1200 \mathrm{~Pa}$ when the flow rate was $50 \mathrm{~cm}^3 / \mathrm{s}$.
(a) Calculate the apparent viscosity and the apparent rate of shear under this condition of flow.
(b) If the same fluid flowing at the rate of $100 \mathrm{~cm}^3 / \mathrm{s}$ through a viscometer tube $0.75 \mathrm{~cm}$ in diameter and $20 \mathrm{~cm}$ long exhibits a pressure drop of $1300 \mathrm{~Pa}$, calculate the flow behavior and consistency indices. Assume wall effects are negligible.

Averell Hause
Averell Hause
Carnegie Mellon University
01:33

Problem 16

Apparent viscosities in centipoises (cP) of $4000,2500,1250$, and 850 were reported on a fluid at rotational speeds of $2,4,10$, and $20 \mathrm{rev} / \mathrm{min}$. This same fluid was reported to require a torque of 900 dyne $\mathrm{cm}$ to rotate a cylindrical spindle $1 \mathrm{~cm}$ in diameter and $5 \mathrm{~cm}$ high within the fluid at $20 \mathrm{rev} / \mathrm{min}$.
(a) Calculate the flow behavior and consistency indices for this fluid.
(b) When exhibiting an apparent viscosity of $4000 \mathrm{cP}$, what would have been the shear rate under which the measurement was made? (FIGURE CAN'T COPY)

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
05:49

Problem 17

The flow behavior and consistency indices of whey at a solids content of $24 \%$ has been reported to be 0.94 and $4.46 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}^{\mathrm{n}}$, respectively. If a rotational viscometer having a full scale torque of 673.7 dyne Acm is to be used for testing the flow behavior of this fluid, determine the diameter and height of a cylindrical spindle to be used such that at the slowest speed of $2 \mathrm{rev} / \mathrm{min}$. The minimum torque will be $10 \%$ of the full scale reading of the instrument. Assume a length to diameter ratio of 3 for the spindle. Would this same spindle induce a torque within the range of the instrument at $20 \mathrm{rev} / \mathrm{min}$ ?

Chai Santi
Chai Santi
Numerade Educator
00:37

Problem 18

Egg whites having a consistency index of $2.2 \mathrm{~Pa} \cdot \mathrm{s}^{\mathrm{n}}$ and a flow behavior index of 0.62 must be pumped through a 1.5 -in. sanitary pipe at a flow rate, which would induce a shear rate at the wall of $150 / \mathrm{s}$. Calculate the rate of flow in $\mathrm{L} / \mathrm{min}$, and the pressure drop due to fluid flow resistance under these conditions.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:17

Problem 19

The system shown in Fig. 6.34 is used to control the viscosity of a batter formulation used on a breading machine. A float indicates the fluid level in the reservoir, and by attaching the float to an appropriate transducer, addition of dry ingredients and water into the mixing tank may be regulated to maintain the batter consistency. The appropriate fluid level in the reservoir may be maintained when a different consistency of the batter is required, by changing the length of the pipe draining the reservoir. If the fluid is Newtonian with a viscosity of $100 \mathrm{cP}$, and if fluid density is $1004 \mathrm{~kg} / \mathrm{m}^3$, calculate the feed rate that must be metered into the reservoir to maintain the level shown. Assume entrance loss is negligible compared to fluid resistance through the drain pipe. (FIGURE CAN'T COPY)

Mahnoor Amin
Mahnoor Amin
Numerade Educator
03:17

Problem 20

The system shown in Fig. 6.35 has been reported to be used for disintegrating wood chips in the pulp industry, after digestion. It is desired to test the feasibility of using the same system on a starchy root crop such as cassava or sweet potatoes, to disrupt starch granules for easier hydrolysis with enzymes to produce sugars for alcoholic fermentation. In simulating the system on a small scale, two parameters are of importance: the shear rate of the slurry as it passes through the discharge pipe and the impact force of the fluid against the plate. Assume that entrance loss from the tank to the discharge pipe is negligible. Under the conditions shown, calculate the shear rate through the drain pipe, and the impact force against the plate at the time the drain pipe is first opened. The slurry has flow behavior and consistency indices of 0.7 and $0.8 \mathrm{~Pa} \cdot \mathrm{s}^{\mathrm{n}}$, and a density of $1042 \mathrm{~kg} / \mathrm{m}^3$.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
03:22

Problem 21

In a falling film direct contact steam heater for sterilization, milk is pumped into a header that distributes the liquid to several vertical pipes, and the liquid flows as a film in laminar flow down the pipe. If the fluid has a density of $998 \mathrm{~kg} / \mathrm{m}^3$, and a viscosity of $1.5 \mathrm{cP}$, calculate the flow rate down the outside surface of each of $3.7-\mathrm{cm}$ outside diameter pipes in order that the fluid film will flow at a Reynolds number of 500 . Calculate the fluid film thickness when flow develops at this Reynolds number.

Ronald Prasad
Ronald Prasad
Numerade Educator
01:32

Problem 22

A sauce product is being formulated to match a reference product (Product A) that has a consistency index of $12 \mathrm{~Pa} \mathrm{~s}^{\mathrm{n}}$ and a flow behavior index of 0.55 . Rheological measurements of the formulated product (Product B) on a wide gap rotational viscometer using a cylindrical spindle $1 \mathrm{~cm}$ in diameter and $5 \mathrm{~cm}$ long are as follows, with speed in rev/min and torque in \% of full scale, respectively: 2,$11 ; 4,18 ; 10,34 ; 20,56$. The viscometer constant is 7187 dyne cm. Calculate the apparent viscosity of Product A and Product B at $0.5 \mathrm{rev} / \mathrm{min}$. At this rotational speed, did the apparent viscosity of Product B match that of Product A?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
12:16

Problem 23

An FMC de-aerator $2.97 \mathrm{~m}$ high and $96.5 \mathrm{~cm}$ in diameter is rated to de-aerate from 4.2 to 8.4 $\mathrm{kg} / \mathrm{s}$ of product. If the product has a density of $1008 \mathrm{~kg} / \mathrm{m}^3$, and has a flow behavior index of 0.44 and a consistency index of $8.1 \mathrm{~Pa} \mathrm{~s}^{\mathrm{n}}$, calculate the film thickness and film velocity to achieve the mid-range $(6.3 \mathrm{~kg} / \mathrm{s})$ of the specified capacity. If half the de-aerator height is to be covered by the fluid film, calculate the time available for any gas bubbles to leave the film into the vapor space in the chamber.

Chareen Guzman
Chareen Guzman
Numerade Educator