12 FORCE, PRESSURE, AND HEAD CALCULATIONS

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65



1 lb of water



2.31 ft



0.433 lb of water



1 ft



1 sq in. AREA



Figure 2.10



1 sq in. AREA



1 ft water = 0.433 psi



1 psi = 2.31 ft water



The relationship between pressure and head.



Pressure (psi) = 0.433 × Head (ft)



(2.12)



Head (ft) = 2.31 × Pressure (psi)



(2.13)



2.12.2 Head

Head is the vertical distance that water must be lifted from the supply tank or unit process to the

discharge. The total head includes the vertical distance the liquid must be lifted (static head); the

loss to friction (friction head); and the energy required to maintain the desired velocity (velocity

head).

Total Head = Static Head + Friction Head + Velocity Head

2.12.2.1



(2.14)



Static Head



Static head is the actual vertical distance the liquid must be lifted.

Static Head = Discharge Elevation − Supply Elevation



(2.15)



Example 2.54

Problem:

The supply tank is located at elevation 108 ft. The discharge point is at elevation 205 ft. What

is the static head in feet?

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Solution:

Static Head, ft = 205 ft − 108 ft = 97 ft

2.12.2.2



Friction Head



Friction head is the equivalent distance of the energy needed to overcome friction. Engineering

references include tables showing the equivalent vertical distance for various sizes and types of

pipes, fittings, and valves. The total friction head is the sum of the equivalent vertical distances for

each component.

Friction Head, ft = Energy Losses due to Friction

2.12.2.3



(2.16)



Velocity Head



Velocity head is the equivalent distance of the energy consumed in achieving and maintaining the

desired velocity in the system.

Velocity Head, ft = Energy Losses to Maintain Velocity

2.12.2.4



Total Dynamic Head (Total System Head)

Total Head = Static Head + Friction Head + Velocity Head



2.12.2.5



(2.17)



(2.18)



Pressure/Head



The pressure exerted by water/wastewater is directly proportional to its depth or head in the pipe,

tank, or channel. If the pressure is known, the equivalent head can be calculated.

Head, ft = Pressure, psi × 2.31 ft/psi



(2.19)



Example 2.55

Problem:

The pressure gauge on the discharge line from the influent pump reads 75.3 psi. What is the

equivalent head in feet?

Solution:

Head, ft = 75.3 × 2.31 ft/psi = 173.9 ft

2.12.2.6



Head/Pressure



If the head is known, the equivalent pressure can be calculated by:



Pressure, psi



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Head, ft

2.31 ft/psi



(2.20)



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67



Example 2.56

Problem:

The tank is 15 ft deep. What is the pressure in psi at the bottom of the tank when it is filled

with wastewater?

Solution:

Pressure, psi



15 ft

= 6.49 psi

2.31 ft/psi



Before we look at a few example problems, we review the key points related to force, pressure,

and head:

• By definition, water weighs 62.4 lb/ft3.

• The surface of any one side of the cube contains 144 in.2 (12 in. × 12 in. = 144 in.2). Therefore,

the cube contains 144 columns of water 1 ft tall and 1 in. square.

• The weight of each of these pieces can be determined by dividing the weight of the water in the

cube by the number of square inches.



Weight =



62.4 lbs

= 0.433 lb/in.2 or 0.433 psi

144 in 2



• Because this is the weight of one column of water 1 ft tall, the true expression would be 0.433

lb/in.2/ft of head or 0.433 psi/ft.

Key point: 1 ft of head = 0.433 psi.



In addition to remembering the important parameter, 1 ft of head = 0.433 psi, it is necessary

to understand the relationship between pressure and feet of head — in other words, how many feet

of head 1 psi represents. This is determined by dividing 1 by 0.433:

Feet of head =



1 ft

= 2.31 ft/psi

0.433 psi



If a pressure gauge reads 12 psi, the height of the water necessary to represent this pressure is 12

psi × 2.31 ft/psi =27.7 ft.

Key point: Both the preceding conversions are commonly used in water/wastewater treatment

calculations. However, the most accurate conversion is 1 ft = 0.433 psi. We use this conversion

throughout this text.



Example 2.57

Problem:

Convert 40 psi to feet head.

Solution:

40 psi

ft

×

= 92.4 ft

1

0.433 psi



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Example 2.58

Problem:

Convert 40 ft to pounds per square inch.

Solution:



40



ft 0.433 psi

×

= 17.32 psi

1

1 ft



As the preceding examples demonstrate, when attempting to convert pounds per square inch

to feet, we divide by 0.433, and when attempting to convert feet to pounds per square inch, we

multiply by 0.433. The preceding process can be most helpful in clearing up the confusion on

whether to multiply or divide. Another way, however, may be more beneficial and easier for many

operators to use. Notice that the relationship between pounds per square inch and feet is almost

two to one. It takes slightly more than 2 ft to make 1 psi. Therefore, when looking at a problem

in which the data are in pressure, the result should be in feet and the answer will be at least twice

as large as the starting number. For instance, if the pressure were 25 psi, we intuitively know that

the head is over 50 ft. We divide by 0.433 to obtain the correct answer.

Example 2.59

Problem:

Convert a pressure of 45 psi to feet of head.

Solution:



45



psi

1 ft

×

= 104 ft

1 0.433 psi



15



psi

1 ft

×

= 34.6 ft

1 0.433 psi



Example 2.60

Problem:

Convert 15 psi to feet.

Solution:



Example 2.61

Problem:

Between the top of a reservoir and the watering point, the elevation is 125 ft. What will the

static pressure be at the watering point?



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Solution:



125



psi

1 ft

×

= 288.7 ft

1 0.433 psi



Example 2.62

Problem:

Find the pressure (pounds per square inch) in a 12-ft deep tank at a point 5 ft below the water

surface.

Solution:

Pressure (psi) = 0.433 × 5 ft

= 2.17 psi

Example 2.63

Problem:

A pressure gauge at the bottom of a tank reads 12.2 psi. How deep is the water in the tank?

Solution:

Head (ft) = 2.31 × 12.2 psi

28.2 ft

Example 2.64

Problem:

What is the pressure (static pressure) 4 miles beneath the ocean surface?

Solution:

Change miles to feet and then to pounds per square inch.

5280 ft/mile × 4 = 21,120 ft

21,120 ft

= 9143 psi

2.31 ft/psi

Example 2.65

Problem:

A 150-ft diameter cylindrical tank contains 2.0 MG water. What is the water depth? At what

pressure would a gauge at the bottom read in pounds per square inch?



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Solution:

Step 1. Change MG to cubic feet:



2,000,000 gal

= 267,380 cu ft

7.48

Step 2. Using volume, solve for depth:



Volume = .785 × D 2 × depth

267,380 cu ft = .785 × (150)2 × depth

Depth = 15.1 ft

Example 2.66

Problem:

The pressure in a pipe is 70 psi. What is the pressure in feet of water? What is the pressure in

pounds per square foot?

Solution:

Step 1. Convert pressure to feet of water:



70 psi × 2.31 ft/psi = 161.7 ft of water

Step 2. Convert pounds per square inch to pounds per square foot:



70 psi × 144 in.2 /ft 2 = 10,080 psf

Example 2.67

Problem:

The pressure in a pipeline is 6476 psf. What is the head on the pipe?

Solution:

Head on pipe = ft of pressure

Pressure = Weight × Height

6476 psf = 62.4 lbs/ft 3 × height

Height = 104 ft

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2.13



71



REVIEW OF ADVANCED ALGEBRA KEY TERMS AND CONCEPTS



Advanced algebraic operations (linear, linear differential, and ordinary differential equations) have

in recent years become an essential part of the mathematical background required by environmental

engineers, among others. Although we do not intend to provide complete coverage of the topics

(engineers are normally well grounded in these critical foundational areas), we review the key terms

and concepts germane to the topics.

Key definitions include:

Algebraic multiplicity of an eigenvalue: the algebraic multiplicity of an eigenvalue c of a matrix A is

the number of times the factor (t – c) occurs in the characteristic polynomial of A.

Basis for a subspace: a basis for a subspace W is a set of vectors {v1, …, vk} in which W is such that:

{v1, …, vk} is linearly independent

{v1, …, vk} spans W.

Characteristic polynomial of a matrix: the characteristic polynomial of a n-by-n matrix A is the

polynomial in t given by the formula det(A – tI).

Column space of a matrix: the subspace spanned by the columns of the matrix considered as a set of

vectors (also see row space).

Consistent linear system: a system of linear equations is consistent if it has at least one solution.

Defective matrix: a matrix A is defective if A has an eigenvalue whose geometric multiplicity is less

than its algebraic multiplicity.

Diagonalizable matrix: a matrix is diagonalizable if it is similar to a diagonal matrix.

Dimension of a subspace: the dimension of a subspace W is the number of vectors in any basis of W.

(If W is the subspace {0}, we say that its dimension is 0.)

Echelon form of a matrix: a matrix is in row echelon form if:

All rows that consist entirely of zeros are grouped together at the bottom of the matrix.

The first (counting left to right) nonzero entry in each nonzero row appears in a column to the

right of the first nonzero entry in the preceding row (if there is a preceding row).

Eigenspace of a matrix: the eigenspace associated with the eigenvalue c of a matrix A is the null space

of A – cl.

Eigenvalue of a matrix: an eigenvalue of a matrix A is a scalar c in which Ax = cx holds for some

nonzero vector x.

Eigenvector of a matrix: an eigenvector of a square matrix A is a nonzero vector x in which Ax = cx

holds for some scalar c.

Elementary matrix: a matrix that is obtained by performing an elementary row operation on an identity

matrix.

Equivalent linear systems: two systems of linear equations in n unknowns are equivalent if they have

the same set of solutions.

Geometric multiplicity of an eigenvalue: when an eigenvalue c of a matrix A is the dimension of the

eigenspace of c.

Homogeneous linear system: a system of linear equations Ax = b is homogeneous if b = 0.

Inconsistent linear system: a system of linear equations with no solutions.

Inverse of a matrix: the matrix B is an inverse for the matrix A if AB = BA = 1.

Invertible matrix: a matrix is invertible if it has no inverse.

Least squares solution of a linear system: a solution to a system of linear equations Ax = b is a vector

x that minimizes the length of the vector Ax – b.

Linear combination of vectors: a vector v is a linear combination of the vectors v1, …, vk if there exist

scalars a1, …, ak in which v = a1v1 + … + akvk.

Linear dependence relation for a set of vectors: a relation for the set of vectors {v1, …, vk} is an

equation of the form a1v1 + … + akvk = 0, where the scalars a1, …, ak are zero.

Linearly dependent set of vectors: the set of vectors {v1, …, vk} is linearly dependent if the equation

a1v1 + … + akvk = 0 has a solution where not all the scalars a1, …, ak are zero (i.e., if {v1, …, vk}

satisfies a linear dependence relation).

Linearly independent set of vectors: the set of vectors {v1, …, vk} is linearly independent if the only

solution to the equation a1v1 + … + akvk = 0 is the solution where all the scalars a1, …, ak are zero

(i.e., if {v1, …, vk} does not satisfy any linear dependence relation).

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Linear transformation: a transformation from V to W is a function T from V to W in which:

T(u + v) = T(u) + T(v) for all vectors u and v in V.

T(av) = aT(v) for all vectors v in V and all scalars a.

Nonsingular matrix: a square matrix A is nonsingular if the only solution to the equation Ax = 0 is x = 0.

Null space of a matrix: the null space of an m by n matrix A is the set of all vectors x in Rn such that

Ax = 0.

Null space of a linear transformation: for a linear transformation, T is the set of vectors v in its domain

such that T(v) = 0.

Nullity of a matrix: the dimension of its null space.

Nullity of a linear transformation: the dimension of its null space.

Orthogonal complement of a subspace: the orthogonal complement of a subspace S of Rn is the set of

all vectors v in Rn so that v is orthogonal to every vector in S.

Orthogonal set of vectors: a set of vectors in Rn is orthogonal if the product of any two of them is 0.

Orthogonal matrix: a matrix A is orthogonal if A is invertible and its inverse equals its transpose; i.e.,

A–1 = AT.

Orthogonal linear transformation: a linear transformation T from V to W is orthogonal if T(v) has the

same length as v for all vectors v in V.

Orthonormal set of vectors: a set of vectors in Rn is orthonormal if it is an orthogonal set and each

vector has length 1.

Range of a linear transformation: the range of a linear transformation T is the set of all vectors T(v),

where v is any vector in its domain.

Rank of a matrix: the rank of a matrix A is the number of nonzero rows in the reduced row echelon

form of A, i.e., the dimension of the row space of A.

Rank of a linear transformation: the rank of a linear transformation (and thus of any matrix regarded

as a linear transformation) is the dimension of its range. Note that a theorem tells us that the two

definitions of rank of a matrix are equivalent.

Reduced row echelon form of a matrix: a matrix is in reduced row echelon form if:

The matrix is in row echelon form.

The first nonzero entry in each nonzero row is the number 1.

The first nonzero entry in each nonzero row is the only nonzero entry in its column.

Row equivalent matrices: two matrices are row equivalent if one can be obtained from the other by a

sequence of elementary row operations.

Row operations: elementary row operations performed on a matrix:

Interchange two rows

Multiply a row by a nonzero scalar

Add a constant multiple of one row to another

Row space of a matrix: the row space of a matrix is the subspace spanned by the rows of the matrix

considered as a set of vectors.

Similar matrices: matrices A and B are similar if a square invertible matrix S is an equivalent to S–1AS = B.

Singular matrix: a square matrix A is singular if the equation Ax = 0 has a nonzero solution for x.

Span of a set of vectors: the span of the set of vectors {v1, …, vk} is the subspace V consisting of all

linear combinations of v1, …, vk. One also says that the subspace V is spanned by the set of vectors

{v1, …, vk} and that this set of vectors spans V.

Subspace: a subset W of Rn is a subspace of Rn if:

The zero vector is in W.

x + y is in W whenever x and y are in W.

ax is in W whenever x is in W and a is any scalar.

Symmetric matrix: a matrix A is symmetric if it equals its transpose; i.e., A = AT.



© 2005 by CRC Press LLC