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Electrical Energy Systems > Equations > Flashcards

Equations Flashcards

Learn renewable energy system equations

1
Q

(General) Convert 1kWh to Joules

A

1kWh = 3,600 seconds x 1000W
= 3.6 x 10^6 J
= 3.6 MJ

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2
Q

(General) Give the equation for work with respect to power and time.

A

W = Pt

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3
Q

(General) State the equation for capacity factor

A

Capacity Factor = Actual output energy / Maximum possible output

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4
Q

(Hydropower) State the relationship between mass, density and volume

A

M = ρV

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5
Q

(Hydropower) Give the equation for stored potential energy

A

PE = ρVgH

where g = 9.81 m/s^2

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6
Q

(Hydropower) Give the equation for input power

A

P = ρQgH

where g = 9.81 m/s^2

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7
Q

(Hydropower) Give the equation for output power

A

P = ηρQgH

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8
Q

(Hydropower) State the equation for Volume Flow Rate in a Pelton turbine

A

Q = A x √(2gH)

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9
Q

(Hydropower) Derive the equation for Power in a Pelton turbine

A

PE = MgH and KE = 1/2 Mv^2

1/2 Mv^2 = MgH

For Potential Energy:
M = ρV so:
E = ρVgH

E = P/t so:

P = (ρVgH)/t

But Q = V/t so:
P = ρQgH

For Kinetic Energy:
KE = 1/2 Mv^2 so:
v = √(2gH)

Therefore:
Q = Av becomes Q = A x √(2gH)

P = ρQgH and Q = A x √(2gH)

So Final Equation is:
P = A x √(2gH) x ρgH

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10
Q

(Hydropower) Give the equation for efficiency in pumps

A

η = (ρQgH) / (Input Power)

Note: Output energy and power are the same for pumps as the input power and energy of the hydropower i.e P = ρQgH and E = ρVgH

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11
Q

(PV Cells) What is the value of 1eV in joules?

A

1eV = 1.602 x 10^-19

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12
Q

(PV Cells) Derive the equation for an ideal current source with a parallel diode

A

Diode current:
I(Diode) = I(Output) x (e^(qV/KT) - 1)

Total Current:
I = I(Short Circuit) - I(Diode)

so:
I = I(Short Circuit) - I(Output) x (e^(qV/KT) - 1)

where:
q is electric charge (1.602 x 10^-19 Coulombs)
K is Boltzmann’s constant (1.38 x 10^-23)
T is temperature in Kelvin

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13
Q

(PV Cells) Convert Celsius to Kelvin

A

Add 273 to the Celsius value to convert it to Kelvin

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14
Q

(PV Cells) Give the equations for the following:

1) Ideal current source with parallel diode and parallel resistor
2) Ideal current source with parallel diode and series resistor
3) Ideal current source with parallel diode, series resistor and parallel resistor
4) Source voltage

A

1) I = (I(Short Circuit) - I(Diode)) - (V/R(Parallel))
2) I = I(Short Circuit) - I(Output)(e^(q(IR(Series)+V)/KT) - 1)
3) I = I(Short Circuit) - I(Output)(e^38.9V(Diode) - 1) - (V(Diode)/R(Parallel))
4) V = V(Diode) - IR(Source)

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15
Q

(PV Cells) Give the equation for theoretical output power

A

P(Theoretical) = V(Open Circuit) x I(Short Circuit)

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16
Q

(PV Cells) State the equation for Fill Factor

A

FF = P(Max) / P(Theoretical) = I(Max Point)V(Max Point) / V(Open Circuit)I(Short Circuit) x 100%

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17
Q

(PV Cells) Give equations for the following:

1) Number of cells in module
2) Number of parallel branches
3) Number of cells in series

A

1) Z = P(Theoretical) / P(Cell)

where Z is the number of cells and P(Cell) is calculatedby multiplying the cell’s Short Circuit Current by Open Circuit Voltage

2) N(Parallel) = I(Short Circuit) / Cell Short Circuit Current
3) N(Series) = V(Open Circuit) / Cell Open Circuit Voltage

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18
Q

(PV Cells) Give the equation for input power of a module

A

P(In) = aG
where a is the area of the module (Metres squared)
G is the insolation (Watts per metre squared)

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19
Q

(PV Cells) State the equation for module efficiency

A

η = P(Max) / P(In) x 100%

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20
Q

(Tidal) State the two equations for energy

A 
E = 1/2 ρAgR^2
E = 1/2 MgR

where R is the range, i.e height difference between an ebb and flow tide

21
Q

(Tidal) Give the equation for mass

A

M = ρAR

22
Q

(Tidal) Convert two tidal cycles into seconds

A

24.8 x 60 x 60 = 8.928 x 10^4s

23
Q

(Tidal) State the equation for the maximum average power

A

P(Max) = (ρAgR^2) / (24.8 x 60 x 60)

If the turbine is two-way, double the answer!

24
Q

(Tidal) Give the power equation for a tidal stream system

A

P = 1/2 ρAv^3

where v is velocity

25
Q

(General) Give the value of pressure for:

1) Air
2) Water

A

1) ρ = 1.29 kgm^-3

2) ρ = 1020 kgm^-3

26
Q

(Wind) State the equation for power

A

P = 1/2 ρAv^3

Note: Exactly the same as tidal stream system

27
Q

(Wind) Give the equation for available power

A

P = 1/2 ρAv^3 x Power Coefficient

28
Q

(Wind) State the equation for power coefficiency

A

C(Power) = Turbine Mechanical Output Power / Theoretical Power of Wind

29
Q

(Wind) Give the equation for annual electricity production

A

Annual Electricity Production = Kv(Mean)^3 AT

where K = 3.2
v(Mean) = annual wind speed
A = Area
T = number of turbines

30
Q

(Wind) State the equation for synchronous speed

A

n(Synchronous) = Frequency / pole pairs

31
Q

(Wind) Give the equation for rotor slip

A

s = (N(Stator) - N(Rotor)) / N(Stator)

Note: Rotor Speed < Stator Speed for motor
Rotor Speed > Stator Speed for generator
FOR A GENERATOR, THE SLIP IS ALWAYS NEGATIVE!!!!

32
Q

(Finance) State the equation for electricity generated per year (kWh)

A

Electricity generated per year (kWh) = Capacity Factor x Rating (kW) x 365 days x 24 hours

Note: The question may give the run-time per year instead, so in this case replace 24 x 365 with the question’s numbers

33
Q

(Finance) Give the two equations for cost per kWh generated

A

1) Cost per kWh generated = Annual Capital Repayment / Average annual energy output
2) Cost per kWh generated = capital cost per kWh + running cost per kWh + fuel cost per kWh

34
Q

(Finance) State the equation that gives payback period

A

Payback Period = Investment Required / Net annual cash inflow

35
Q

(Finance) Give the equation for annuity

A

A = (rV)/(1-(1+r)^-n)

where A = Annual Capital Cost, r is the discount or interest rate, V is the Total Capital Cost and n is the number of years

36
Q

(Finance) State the equation for fuel costs

A

Fuel cost per kWh generated = (Cost in pence / Energy in kWh) / (Efficiency)

37
Q

(Hydropower) Give the names of three types of turbines and their properties

A

Pelton: High head, low flow rate
Francis: Medium head, covers largest variation of heads making them the most common turbine
Kaplan: Low head, large volume flows

38
Q

(Hydropower) Give the names of two types of pumps and their properties

A

Centrifugal: Centrifugal force pushes water out radially

Axial Flow: Water enters and exits along the pipe in the same direction

39
Q

(PV Cells) What does MPPT stand for and what does it do?

A

MPPT (Maximum Power Point Tracker) tracks the locus of the maximum power point and forces
the circuit to operate at its MPP at all times for maximum efficiency
It changes the dc voltage up or down so the maximum power is drawn from the PV array based on
the PV curve. This can be done with a buck-boost converter.

40
Q

(PV Cells) How does temperature affect PV performance?

A

Temperature decreases Voc and increases Isc only slightly, so adversely affects PV performance

41
Q

(Wind) State an advantage and disadvantage of both horizontal and vertical turbines

A

– Horizontal –
Advantage - Higher in air so catches faster wind speeds
Disadvantage - The generator and gearbox must be placed high up so are hard to reach for
maintenance

– Vertical –
Advantage - Can harness winds from any direction without need of yaw control
Disadvantage - Close to ground, slower speeds, less power

42
Q

(Wind) What is the purpose of Yaw Control and Pitch Control in a turbine?

A

Yaw Control: Makes sure turbine faces wind for maximum efficiency
Pitch Control: Sets turbine blades at optimal angle

43
Q

(Wind) State an advantage and disadvantage of upwind and downwind turbines

A

– Upwind –
Advantage - Operate more smoothly than downwind
Disadvantage - Require complex yaw control systems

– Downwind –
Advantage - Wind controls the yaw for it
Disadvantage - When the blade moves behind the tower it experiences reduced wind for an instance that causes the blade to flex, this can damage the blade and increases blade noise which in turn reduces power output

44
Q

(Wind) Derive the equation for input power to a turbine

A 
E = 1/2 mv^2
But P = E/t so
P = 1/2 m/t v^2
Also m = ρV
P = 1/2 ρV/t v^2
Now Q = V/t
P = 1/2 ρQv^2
But Q = Av
P = 1/2 ρAv^3
45
Q

(Finance) Give the equation for capacity factor in terms of hours per year

A

Capacity Factor = Run-time per year / 365 x 24

46
Q

(Finance) State the equation for the Total Capital Cost

A

Total Capital Cost = Power Generation in kW x Cost per kW

47
Q

(Wind) Sketch a typical wind speed-power curve of a wind turbine and mark the three crucial parameters on it

A

The crucial parameters are from left to right: Cut-in speed, rated speed, shut-down speed. The graph you will have to remember yourself!

48
Q

(Tidal) Describe the principle of tidal stream systems for generating electric power and give one advantage and disadvantage in their use

A

Tidal stream systems collect the energy from the horizontal movement of the tides. A flood
current moves water closer to the shore, whereas ebb pushes water away. Tidal streams use
turbines similar to those of wind to collect this energy. This has to be built close to the
shoreline as the tidal currents have very little effect in the open ocean.
Advantage - Costs far less than a lagoon system
Disadvantage - Has to be built close to the shore, so could be dangerous for both humans and for wildlife

Electrical Energy Systems (2 decks)

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