Gay E. Canough Solar Math. PV Installer s Class: Units and Math

Transcription

1 Gay E. Canough 1.1--Solar Math October 2012 PV Installer s Class: Units and Math 1

2 Units Electricity Fluid KW, a unit of power Horsepower, a unit of power KWh, a unit of energy BTU, a unit of energy Voltage --- Volts Pressure --- Pounds per square inch (psi), feet of water (head), kilopascals (International unit) Current --- Amps Flow --- Gallons per minute, liters per minute Charge in a battery --- volts, specific gravity of electrolyte Volume in a cistern --- Gallons, liters, cubic feet, cubic centimeters October 2012 PV Installer s Class: Units and Math 2

3 Math symbols reminder 4 ways to show division: and 12/4 1) 2) number A / number B means number A number B that is number A divided by number B 3) number A number B means number A number B that is number A divided by number B 4) In some math texts division is shown as number B divided into number A with the long division sign like this: number B ) number A which is the same as number A number B October 2012 PV Installer s Class: Units and Math 3

4 Multiplication is commutative which means it can be done in any order. For example 3 x 4 x15 = 15 x 3 x 4 communicative is something different... it means 1. Inclined to communicate readily; talkative. 2. Of or relating to communication. October 2012 PV Installer s Class: Units and Math 4

5 Solar Powered Algebra The most sophisticated algebra we do involves only 2 rules: 1) Anything divided by itself = 1 2) Whatever operation is performed on one side of an equation must be done to the other. The very word equation means equal. In politics this has similar meaning. In the USA, people of all races have the same (equal) rights. October 2012 PV Installer s Class: Units and Math 5

6 Converting units FACTOR-LABEL 1 kwh sq.meter 1 Ohm Kilo-foot x 1 sq.meter = 1 kwh x 1000 feet = 1 Ohm 1 kwh x 3412 BTU x 1 kwh 1 Therm = 100,000 BTU Therms October 2012 PV Installer s Class: Units and Math 6

7 Practice 1) Convert kwh/sq. meter /day to BTUs/sq.foot/year There are sq. ft. per sq. meter and 3412 BTUs per kwh 2) Convert Square Meters of PV to kw STC of PV Cell efficiency= 13.5%, Cell size = 4.5, number of cells in module = 36 Module size = 19.7 x 43.6 Given that the area of the module is larger than the area of the solar cells, what is the Watt/square foot output of the module? October 2012 PV Installer s Class: Units and Math 7

8 A Kilowatt is a unit of Power. 1 Kilowatt = 1,000 Watts Units A Kilowatt Is a Unit of Power 1 Watt = 1 Joule per second. A Joule is a tiny unit of energy measured by British physicist Joule in the 1800 s. He used small metric weights, force and distances to define one joule. Horsepower is also a unit of Power. One horsepower is the amount of power needed to move 550 foot-pounds per second. One horsepower = 746 watts October 2012 PV Installer s Class: Units and Math 8

9 Units How Much is a Kilowatt-Hour? TEN 100 W lightbulbs left on for 1 hour use: 10 bulbs x 100 Watts x 1 hr = 1000 WATT-HOURS = 1 KILOWATT-HOUR = 1 kwh 1 hair dryer that uses 1000 Watts left on for 1 hour uses: 1 dryer x 1000 W x 1 hour = 1000 WATT-HOURS = 1 KILOWATT-HOUR = 1 kwh While bicycling a 150 pound person will use 20 kilojoules per minute. To expend 1 kwh, you will be pedaling about 3 hours. October 2012 PV Installer s Class: Units and Math 9

10 Watts = Amps (I) x Volts (V) Units Speaking of Watts To calculate how much energy a customer needs, find the Wattage of each appliance and multiply it times the number of hours the appliance is used per day.if the Watts are not given but the Amps are, use the formulas above to calculate Wattage. Add up the Watt hours of all appliances. Divide this sum by 1,000 to convert the number to KWh (because 1 KW = 1,000 W). Total energy needs = the sum of Watts x hours = Watt hours 1,000 = KWh October 2012 PV Installer s Class: Units and Math 10

11 Example Well pump runs on 240 V and takes 9 Amps. 240 x 9 = 2160 W or kw If it runs for 2 hours, it consumes 2.16 x 2 = 4.32 kwh October 2012 PV Installer s Class: Units and Math 11

12 Example AC in 120 V 30 A 120 V at 30 A = 24 V at? Use I x V 120 x 30 = 24 x? Solve:?= (120 x 30)/24 = 150 A DC out 24 V 150 A October 2012 PV Installer s Class: Units and Math 12

13 The rules you will use the most: Algebra review When solving for an unknown in an equation, you can do so by repeatedly applying these rules: 1) Whatever operation you do on the right-side of an equation, you must also do to the left-side in order to maintain equality. 2) Anything divided by itself is one (1). October 2012 PV Installer s Class: Units and Math 13

14 Example Distance = speed x time You drive 100 miles at 65 mph. How long does it take? Solve for time: Distance speed x time = = time Speed speed Why did I cross out speed? Because anything divided by itself = 1 And 1 x thing = thing. October 2012 PV Installer s Class: Units and Math 14

15 You would have to use this to find voltages and currents From I1 x V1 = I2 x V2 Or to find ohms/kf from (0.2 x d x I /V) x (ohm/kf) = %Voltage Drop October 2012 PV Installer s Class: Units and Math 15

16 Figure out a system to Power the whole load like this Formula #1 Your Load in KILOWATT-HOURS per year Divided by Hours of Full Sun x system efficiency If your yearly load is 7000 kwh, the sun in New York City is 1642 kwh/m 2 per year and the system efficiency is 75% (0.75) then the PV system you will need to run it is: 7000/ (1642 x 0.75) = 5.68 kw PV System October 2012 PV Installer s Class: Units and Math 16

17 Total System Performance = Yearly KWh (ac) Produced This is the product of: Total amount of energy falling on solar collector at installed tilt for the year s total sun hours (from NREL data) Size of PV array in total KW Efficiency of the inverter Efficiency of the DC wiring (100% minus percent lost due to voltage drop) 1- loss due to dirt and voltage mismatch. x0.85 for east or west facing roof October 2012 PV Installer s Class: Units and Math 17

18 String Sizing Formulas 1) Find the maximum number in series allowed: Max Voltage INVERTER can stand/(pv module Voc x 1.25) =600 (for the SMA 2500) /(36.1 x 1.25) =13.29 modules, or 13 as the maximum This ensures you never exceed 600 V on the open circuit 2) Find the minimum number in series Lower edge of PPT window/(pv module Vmp x 0.8) =234 (for SMA 2500) /(28.5 x 0.8) = or at least 11 modules as minimum This ensures you stay in the peak power tracking regime when the modules are hot October 2012 PV Installer s Class: Units and Math 18

19 Where: Voltage drop formula (0.2 x d x I /V) x (Ω/kf) = %Voltage Drop I is the circuit current, which for source circuits is usually taken as the maximum power current, Imp d = one way distance in feet V is the voltage at which you want to find VD, and Ω/kft is the wire s resistivity in Ohms per 1000 feet and is found from NEC Chapter 9, Table 8, Conductor Properties. Math note: 2d is the round trip distance. To convert to %, one multiplies by 100, but to convert feet to kilo-feet, one divides by THEREFORE 2 x 100/1000 = 0.2 October 2012 PV Installer s Class: Units and Math 19

20 Calculating Voltage Change with Temperature V(at Temp) = V(at STC) + [Temp coefficient x (T of interest 25 C)] EXAMPLE The Sharp 208 module has a temperature coefficient of V/C That s volts per degree C. It is negative because the voltage gets HIGHER as the temperature gets LOWER. The open circuit voltage is 36.1 V at STC (1000 W/sq. m., 25 C). At a temp of interest of 20 C (-4 F): Voc (at -20) = [ x (-20-25)] = [ x -45) = [5.4] = 41.5V (code: 36.1 x1.25 =45.12V) October 2012 PV Installer s Class: Units and Math 20

21 Other formulas Voltage = Current x Resistance (Ohm s Law) Watt-hours = Amp-hours x voltage Parallel wiring modules: Current Adds, Voltage remains the same Series wiring modules: Voltage adds, Current remains the same October 2012 PV Installer s Class: Units and Math 21

22 Useful conversion Factors 1 square meter = square feet 1 square foot = 144 square inches 1 kw = 1000 Watts 1 kwh = 1000 Wh = 3412 BTUs 1 acre = square feet 1 square mile = square feet October 2012 PV Installer s Class: Units and Math 22

23 Solar Trigonometry: Dealing with Angles Tangent A = opposite / adjacent Sine A = opposite / hypotenuse Cosine A = adjacent / hypotenuse Angle B = Angle A (Rule of Similar Angles) Opposite (Y) B Hypotenuse (H) Angle A Adjacent (X) October 2012 PV Installer s Class: Units and Math 23

24 Slope in degrees = TAN -1 (rise/run) Find the roof slope in degrees for rise/over run of: 4/12 Solar Trigonometry: Dealing with Angles RISE 5/12 6/12 7/12 8/12 RUN Angle (slope) To find the inverse tangent, TAN -1, use a trig table or the TAN -1 or ATAN key on scientific calculator 10/12 12/12 October 2012 PV Installer s Class: Units and Math 24

25 Solar Trigonometry: Dealing with Angles Find a place where there is no shade from 9 A to 4 P. No shade is guaranteed when the object is 3 times as far away as it is tall. Shade problem October 2012 PV Installer s Class: Units and Math 25

26 Solar Trigonometry: Dealing with Angles Up-slope dimension cross check: measure slope, do trig cosine (angle) = adjacent/ hypotenuse so the hypotenuse = adjacent/cosine(angle) hypotenuse angle adjacent October 2012 PV Installer s Class: Units and Math 26

27 I snuck some algebra in there.. I ll fill in the steps: cosine (angle) = adjacent/ hypotenuse multiply BOTH sides of this equation by hypotenuse. This leads to: hypotenuse x cosine (angle) = (adjacent/hypotenuse) x hypotenuse but anything divided by itself is 1, so: adjacent hypotenuse x hypotenuse = adjacent x hypotenuse hypotenuse = adjacent x 1 = adjacent October 2012 PV Installer s Class: Units and Math 27

28 If you have to do tilt-ups on roof, space the rows so one does not shade another. Triangle #1 Triangle #2 October 2012 PV Installer s Class: Units and Math 28

29 Triangle #1 Finding Y, the projected height The hypotenuse is the module length, 2 8 or = 32 sine(48) = Y/ hypotenuse AND SO Y= sine (48) x hypotenuse Y= sine(48) x 32 = x 32 = Triangle #2 Now, we find the distance between rows: 18 deg is the minimum sun angle (December at 10 AM) Tangent(18) = opposite over adjacent = Y/X = 24 /X Solve for X X = 24 /tan(18) = = 6 feet If you are unfamiliar with algebra, make a scale drawing instead. You will need a ruler, protractor and calculator to do this. October 2012 PV Installer s Class: Units and Math 29

30 Tilt-up on NW roof to face PV South October 2012 PV Installer s Class: Units and Math 30