GATE TF flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rank of a matrix

The number of linearly independent rows (or columns). It fixes the dimension of the solution space of A x = 0 as n minus rank(A), and decides consistency of linear systems.

Eigenvalue trace relation

The sum of all eigenvalues of a square matrix equals its trace, the sum of diagonal entries. The product of eigenvalues equals the determinant.

Leibniz integral rule

Differentiating an integral with variable limits gives f(b(x))b′(x) minus f(a(x))a′(x). It handles limits that depend on the differentiation variable.

Second derivative test

At a stationary point, fxx fyy − fxy² positive with fxx positive gives a minimum; positive with fxx negative gives a maximum; negative gives a saddle point.

Integrating factor method

For y′ + P(x)y = Q(x), multiply by exp(∫P dx) to make the left side an exact derivative, then integrate both sides directly.

Newton–Raphson convergence

The iteration x − f(x)/f′(x) converges quadratically near a simple root, roughly doubling correct digits per step, but fails if f′ vanishes at the iterate.

Trapezoidal rule error order

For a smooth integrand, the composite trapezoidal rule error decreases as h squared, where h is the subinterval width; Simpson's 1/3 rule achieves fourth order.

Bayes theorem form

The conditional probability P(A|B) equals P(B|A)P(A) divided by P(B). It updates a prior probability using observed evidence.

Poisson mean and variance

For a Poisson random variable counting rare events, the mean and the variance are both equal to the rate parameter, and the standard deviation is its square root.

Steady-state energy balance

With no accumulation, energy in minus energy out plus generation minus consumption equals zero. Generation covers exothermic reaction heat; consumption covers endothermic heat.

Dimensionally consistent equations

Every additive term in a valid physical equation must carry identical dimensions. Checking dimensions catches errors before any numerical substitution is attempted.

HHV versus LHV

Higher heating value counts latent heat recovered by condensing product water vapour; lower heating value excludes it. The difference equals mass of water formed times latent heat.

Stoichiometric air for coal

Mass of oxygen needed per kg coal is approximately 2.67C + 8H + S − O using mass fractions. Air mass follows by dividing by the oxygen mass fraction in air, about 0.232.

Excess air definition

Actual air supplied divided by stoichiometric air, expressed as a percentage above 100. It raises oxygen in flue gas and lowers CO₂ concentration on a dry basis.

Dry flue gas basis

Orsat-type analysis reports CO₂, O₂, CO and N₂ after water vapour is removed. Volume fractions therefore sum to 100 on a dry basis, not on a wet basis.

Solar declination angle

The angle between the sun's rays and the equatorial plane, approximated as 23.45° sin[360(284 + n)/365] for day number n. It varies between about plus and minus 23.45°.

Hour angle formula

Fifteen degrees multiplied by the difference between local solar time and noon. Negative before noon, positive after, and zero at solar noon.

Solar zenith angle relation

Cosine of the zenith angle equals sin φ sin δ plus cos φ cos δ cos ω, where φ is latitude, δ declination and ω hour angle. It gives sun position for any time.

Collector efficiency curve

Efficiency equals F_R(τα) minus F_R U_L times the difference between inlet fluid and ambient temperature divided by incident irradiance. The plot is a straight line with negative slope.

Betz limit

The maximum fraction of wind power a rotor can extract is 16/27, about 59.3 percent. It follows from momentum theory assuming an ideal, frictionless actuator disc.

Wind power in the stream

Power available in wind crossing area A is half the air density times A times cube of wind speed. Doubling wind speed multiplies available power by eight.

Tip-speed ratio

Blade tip speed divided by free-stream wind speed, equal to angular velocity times rotor radius divided by wind speed. Power coefficient depends strongly on this ratio.

Hydro power equation

Electrical output equals density times gravity times volumetric flow times net head times overall efficiency. Net head subtracts losses in penstock and intake from gross head.

Pelton wheel jet velocity

Ideal jet speed is the square root of twice gravity times head. Maximum energy transfer occurs when bucket speed is roughly half the jet speed.

Capacity factor

Annual energy generated divided by the product of rated power and 8760 hours. It measures how fully a plant uses its nameplate rating over a year.

Levelized cost of electricity

Total discounted lifetime cost divided by total discounted lifetime energy generated. It allows comparison of technologies with different capital costs, fuel costs and lifetimes.

Latent heat of vaporisation

Energy absorbed when unit mass changes from liquid to vapour at constant temperature. For water near 100 °C it is about 2257 kJ per kg.

Adiabatic flame temperature

Temperature reached by products when fuel burns completely with no heat loss to surroundings. It rises with preheated reactants and falls with excess air.

Air-fuel ratio

Mass of air supplied per unit mass of fuel. It is compared with the stoichiometric ratio to define whether the mixture is lean or rich.

Photovoltaic fill factor

Maximum power divided by the product of open-circuit voltage and short-circuit current. Higher fill factor means a squarer I–V curve and better cell quality.

Biomass calorific value

Energy released per unit mass on complete combustion, usually quoted on a dry basis. Moisture content lowers the effective heating value because evaporation consumes heat.

Sensible versus latent storage

Sensible storage raises a material's temperature, with capacity mass times specific heat times temperature rise. Latent storage absorbs heat at phase change with much higher energy density.