HP 20S - Normal Distribution, Direction Cosine, Fire Friction Loss

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 strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Spotlight: Akron Brass FireCalc Pocket Computer

Spotlight: Akron Brass FireCalc Pocket Computer

Welcome to a special Monday Edition of Eddie’s Math and Calculator blog.

This is an instance of a small gamble and the gamble paid off because the calculator works! I found a one of a kind calculator at the Redlands Thrift Store in Redlands, CA: the Akron Brass FireCalc Pocket Computer. The specialty is mathematical calculations in fire fighting. I paid $4.95, and it probably would cost, I guess $40-$50 or higher retail because this is a specialty calculator.

Quick Facts

Model: FireCalc Pocket Computer

Company: Akron Brass

Timeline: ????

Type: Fire Science Solver, 4 Function Calculator

Memory: No conventional memory registers, but input in the solvers are stored in memory

Power: 2 x LR44 batteries

The four function calculator has a square key, [ x^2 ], along with a square root key [ √ ]. There are two clearing keys:

[ CLEAR DISPLAY ]: acts like as clear entry key, which makes the display show 0.

[ ALL CLEAR ]: clears all the registers, and the display shows AKRON.

The display holds room for 8 digits. The display also shows alphabetic characters being built from segments.

Disclaimer: I am not a fire fighter or an expert in fire fighting mathematics. This is another topic to explore. I am going to do the best I can in describing the functions of the calculator.

A Solver for Fire Fighters

The FireCalc Pocket Computer features eight solvers. I’m going to be give a summary of how each solver as described as in the manual (see the manual section below). The solvers are the top two rows of blue keys. U.S. units are used.

There are four rows of gray keys with measurements marked ¾” all the way to 6”. Those are quick entry keys. For example, pressing [ 1 ¾” ] enters 1.75 in one keystroke.

The solvers are:

[ ENGINE PRESSURE ]: calculates the engine pressure in psi (pounds per square inch) of the water hose (I think) given the pressure, flow (gallons per minute), and length of the hose. There is also a prompt for the number of Siamese lines used (defaults to 1).

[ FRICTION LOSS ]: calculates the friction loss of a hose lay given the hose size and length. The loss is shown in psi.

(FLOW RATE) [ STRAIGHT TIP ] or [ FOG NOZZLE ]: Find the flow rate of water through the hose when it is a straight bore tip or fog nozzle in GPM (gallons per minute), respectively. For the fog nozzle, the rated nozzle pressure is skipped (the manual has it prompted).

(REACTION FORCE) [ STRAIGHT TIP ] or [ FOG NOZZLE ]: Find the reaction force in Lbs (pounds fource) of water through the hose when it is a straight bore tip or fog nozzle, respectively.

[ APPLIC. RATE ]: Find the minimum rate of water, in GPM, that is required to extinguish a fire given the rooms’ length, width, and height in feet.

[ RATE OF FLOW ]: I did not see this calculation the manual, but given from the prompts, this calculates the rate of flow of water given the following parameters: nozzle pressure, English pressure, hose size, number of Siamese lines used, number of floors, and hose length.

Because the number of solvers don’t match the manual exactly, I think this model either predates or succeeds the model described in the manual. On the back of the project it states “Cygnus of South Florida Electric Calculator. Project No. 302”. Does that mean anything?

Manual

Manual to the FireCalc from Akron Brass Company, written in 2008. It’s not the exact module that I have, but it’s pretty close: https://smhttp-ssl-61500.nexcesscdn.net/media/pdf/9900_FireCalcInstructions.pdf

Until next time, be safe and wishing you happiness (and safety from fires),

Eddie

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.