HP 20S - Normal Distribution, Direction Cosine, Fire
Friction Loss
HP
20S Normal Distribution
Goal:
To estimate the area of the normal curve
∫(
e^(-x^2 / 2) dx, a, b) / √(2 * π)
Steps:
1.
Enter the programming editor: [ <| ] [ R/S
] {PRGM}
2.
Load the integration program: [ <| ] [ ←
] {LOAD} [ e^x ] { B }. The screen shows “int”.
3.
Go one up one step to get to Step 58: [ <|
] [ 8 ] { ↑ }
4.
Enter the program after 58: 61, 41, F (LBL F)
51, 11
|
x^2
|
45
|
÷
|
2
|
2
|
74
|
=
|
32
|
±
|
12
|
e^x
|
45
|
÷
|
33
|
(
|
2
|
2
|
55
|
×
|
61, 22
|
π
|
34
|
)
|
11
|
√
|
74
|
=
|
5.
Store the lower limit in register 5: a [ STO ] [ 5 ]
6.
Store the upper limit in register 6: b [ STO ] [ 6 ]
7.
Enter the number of intervals, it must be an even integer, and
execute label A: n [ XEQ ] [ √ ] { A }
Example
At n
= 20 intervals, estimate areas (ALL setting)
a =
0, b = 3; area ≈ 0.498649878
a =
-3, b = 3; area ≈ 0.997293118
a =
-1, b = 2; area ≈ 0.818595675
HP
20S: Direction Cosines
The
direction cosines of 3D vector v = [x, y, z] are:
a =
arccos(x / norm(v))
b =
arccos(y / norm(v))
c =
arccos(z / norm(v))
where
norm(v) = √( x^2 + y^2 + z^2 )
The
following program sets the angle mode to degrees, however, a change
in the second step will allow the user to use radians or grads
instead. The program uses the rectangular to polar conversion to
obtain the norm.
Math
note:
Find
the magnitude of (√(x^2 + y^2), z).
magnitude
= √(
[√(x^2 + y^2)]^2 + z^2 )
= √(
x^2 + y^2 + z^2)
=
norm(v)
Executing
the →P command gives the angle first. Obtaining the magnitude
requires a swap. ( [ <| ] [ INPUT ] {SWAP}
).
61, 41, b
|
LBL B
|
61, 23
|
DEG (61, 24 for RAD, 61, 25 for GRD)
|
22, 1
|
RCL 1
|
31
|
INPUT
|
22, 2
|
RCL 2
|
51, 21
|
→ P
|
51, 31
|
SWAP
|
31
|
INPUT
|
22, 3
|
RCL 3
|
51, 21
|
→ P
|
51, 31
|
SWAP
|
21, 4
|
STO 4
|
22, 1
|
RCL 1
|
41, C
|
XEQ C
|
26
|
R/S
|
22, 2
|
RCL 2
|
41, C
|
XEQ C
|
26
|
R/S
|
22, 3
|
RCL 3
|
41, C
|
XEQ C
|
61, 26
|
RTN
|
61, 41, C
|
LBL C (subroutine)
|
45
|
÷
|
22, 4
|
RCL 4
|
74
|
=
|
51, 24
|
ACOS
|
61, 26
|
RTN
|
Store
x in register 1, y in register 2, and z in register 3. The angles
are shown in order
Examples (FIX 4):
x =
4, y = 8, z = 5
Direction
Cosines: a ≈ 67.0231°, b ≈ 38.6734°, c ≈ 60.7941°
x =
-3, y = 8, z = 6
Direction
Cosines: a ≈ 106.6992°, b ≈ 39.9807°, c ≈ 54.9217°
Source:
“Direction
Cosine” Wikipedia. Accessed November 5, 2024.
https://en.wikipedia.org/wiki/Direction_cosine
HP
20S: Determining the Coefficient for Friction Loss
When
fighting fires, the friction loss of a hose lay can be determined by
the formula:
FL =
C * (flow rate/100)^2 * (hose length/100)
where:
C =
coefficient
flow
rate = the rate of water in GPM (gallons per minute)
hose
length = length of the hose in ft (feet)
FL =
friction loss in PSI (pounds per square inch)
This
formula assumes a single line is used.
Solving
for C:
C =
FL / ((flow rate/100)^2 * (hose length/100))
The
friction loss was determined by using various flow rates and hose
lengths by using the FireCalc Pocket Calculator. You can see my
spotlight on the FireCalc Pocket Calculator here:
https://edspi31415.blogspot.com/2024/11/spotlight-akron-brass-firecalc-pocket.html
Friction
Loss Table:
1”
Hose Size
GPM ↓ / Length →
|
100 ft
|
150 ft
|
200 ft
|
100
|
150
|
225
|
300
|
150
|
338
|
506
|
675
|
200
|
600
|
900
|
1200
|
1.5”
Hose Size
GPM ↓ / Length →
|
100 ft
|
150 ft
|
200 ft
|
100
|
24
|
36
|
48
|
150
|
54
|
81
|
108
|
200
|
96
|
144
|
192
|
2”
Hose Size
GPM ↓ / Length →
|
100 ft
|
150 ft
|
200 ft
|
100
|
8
|
12
|
16
|
150
|
18
|
27
|
36
|
200
|
32
|
48
|
64
|
The
coefficient is built in to the FireCalc. I used the HP 20S to
extract the coefficient by the following program:
61, 41, A
|
LBL A
|
33
|
(
|
22, 2
|
RCL 2
|
45
|
÷
|
1
|
1
|
0
|
0
|
0
|
0
|
34
|
)
|
51, 11
|
x^2
|
55
|
×
|
33
|
(
|
22, 3
|
RCL 3
|
45
|
÷
|
1
|
1
|
0
|
0
|
0
|
0
|
34
|
)
|
74
|
‘=
|
15
|
1/x
|
55
|
×
|
22, 1
|
RCL 1
|
74
|
=
|
61, 26
|
RTN
|
Values
are stored in the following registers:
Register
1 = friction loss (PSI)
Register
2 = flow rate (GPM)
Register
3 = hose length (ft)
Fortunately,
running the program with various data points above, I obtain the
coefficient as:
1”
Hose Size: coefficient = 150
1.5”
Hose Size: coefficient = 24
2”
Hose Size: coefficient = 8
Source:
Task
Force Tips. “Hydraulic Calculations Every Firefighting Needs to
Know” Firefighter Trending Report. 2024. Retrieved November
10, 2024.
https://tft.com/hydraulic-calculations-every-firefighter-needs-to-know/
Enjoy!
Eddie
All
original content copyright, © 2011-2025. Edward Shore.
Unauthorized use and/or unauthorized distribution for commercial
purposes without express and written permission from the author is
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