Questions and Answers of The Physics Energy

Consider a house in a northern location that needs a total of 100 GJ of thermal energy over a winter season from November through March. Assume that the house has a ground source heat pump that
Assume that a region of continental crust has a typical surface heat flux of 65 mW/m2, crustal density of 2750 kg/m3, and typical crustal abundances of radioactive nuclides (as given in Table 20.6).
Check eq. (32.4)by estimating the radioactive heat production rate using data from Table 20.6Take the fractional abundances of 235U, 40K to be 0.72%, 0.012%. Take the average energy release in 40K
Make a very rough estimate of the thermal energy content of Earth, assuming that the core has radius 3480 km, temperature 4000 K, density 11000 kg/m3 and heat capacity 800 J/K kg, and that the
Estimate the total power of the Gulf Stream in the Florida Straits, given that the mass flux is roughly 30 Sv. Take the speed of the flow to be 4 m/s.
Consider an idealized tidal reservoir of surface area A = 30 km2 that fills and drains through a narrow channel of width w = 60 m and depth d = 20 m. Assume that the tidal range is h = 2.5 m. For
When a tidal barrage follows the water release profile shown in Figure 31.27 it only captures a fraction of the energy (31.35) available in the water stored at high tide. The discharge is
Consider a tidal barrage constructed to enclose a lagoon of surface area 5 km2, with typical tidal range of 4 m. What is the maximum possible average power output of such a barrage? How much would
A basin under consideration for a tidal barrage power installation has an area that grows quadratically with height z, A(z) = A0z2/h2, where h is the difference between high and low tide. Calculate
Consider building a tidal barrage along a straight section of beach in California. Assume that the average tidal range is 3 m, with two high tides a day. Assume that the tidal basin is 20 m wide and
Is tidal power a renewable resource? Assuming the rate of energy loss due to the tides remains constant at 3.9 TW, estimate the time it will take for Earth to lose 10% of its rotational energy to
Show that energy loss from Earth’s rotational kinetic energy at a rate of 3.9 TW (including dissipation and energy transferred to lunar orbit) corresponds to an increase in length of day of 2.3
Imagine that Earth rotated about its axis once per month and that the Moon moved in a circular orbit above Earth’s equator. Under these tidally locked conditions, the Moon and the resulting tidal
In the JONSWAP model, the energy transferred to the moving ocean surface waters is proportional to the distance over which wind travels (fetch, F). For a steady wind at 10 m/s compute the rate of
A tsunami is a very long-wavelength wave produced by a sudden movement in the sea floor over a distance of hundreds of kilometers. The wavelengths in a tsunami are generally much greater than ocean
Consider a deep-water wave with amplitude a = 1 m and period 12 s. Compute the energy density and power. When the wave moves into shallow water, what is the height and energy density of the wave at 2
Use dimensional analysis to determine the dispersion relation ω(k) for surface tension waves (up to a multiplicative constant), which can only depend on the surface tension σ, the density ρ, and
Show that the quantity has the dimensions of length, where the surface tension σ has units of energy per unit area. Look up the surface tension of water and evaluate l for water. When a small
Estimate the power output possible for a hydroelectric dam with hydraulic head Z = 30 m, flow rate Q = 300 m3/s, and turbine efficiency 90%.
Compare land use of hydropower to solar power by computing the ratio of power output to reservoir surface area for the Three Gorges Dam (maximum capacity 22500 MW, reservoir area 1084 km2) and the
Use blade-element theory to compute the correction to the axial-momentum theory expression for the power (30.7)To first order in cd/cl for an optimized blade. First, recompute f T and f N including
Model the blades of a HAWT rotor as rods of equal length and mass, equally spaced around a circle, rotating with angular frequency Ω. Compute the moment of inertia of the rotor about the vertical
The axial-momentum theory approximation to blade design breaks down when r/R becomes too small. Use eqs. (30.11)and (30.15) to show that the condition KB < 2πr limits the angle φ to sin φ
Axial-momentum analysis led to a power coefficient CP = 4a(1 − a)2 for a HAWT eq. (30.7)Compute the power coefficient another way: start from the tangential force per unit length f T in the simple
Derive eq. (30.15)by equating the normal force per unit length on a blade element given in eq. (30.12)with the thrust per unit length obtained from the axial momentum analysis (30.14). 8πα K sin o
Verify the results of eq. (30.7) and evaluate them at the Betz limit. v4 = (1 – 2a)v1, %3D v2 = (1 – a)v1, 1- a AT, 1- 2a A4 A1 = (1 – a)AT, 1 P = = ;pov{AT4a(1 – a)² .
Wind frequency data obtained at a height of 10 meters at a particular location is summarized by a Weibull distribution with λ = 9 m/s and k = 1.3. The land use is best characterized as “rough
The maximum power of a Savonius wind turbine (see Example 30.2) is claimed to be P = 0.36 hrv3, where P is in watts, and the height h and radius r of the Savonius rotor, and the wind velocity v, are
A variable-speed HAWT has a power coefficient of 0.45 at its rated wind speed vr and maintains its rated power for wind speeds between this value and its cutout speed of 25 m/s. If the rotor diameter
The isothermal compressibility of a fluid is the relative change of volume with pressure at constant temperatureβ = −(1/V)∂V/∂p|T. Qualitatively describe and sketch the compressibility of
A wind pump based on the “American farm windmill” design Figures 30.2(b) and 30.7with rotor diameter 4.5 m and tip-speed ratio λ ∼ 1 claims to be able to pump 2400 L/h of water from a
A modern medium size commercial jet plane has a wing span of L = 30 m and a mean chord length of K = 4 m. Its lift coefficient is cl = 0.83 at its cruising angle of attack α ∼ 2◦. What is the
Investigate the location of the stagnation points for fluid flow around a cylinder as described in §29.4.1. Show that for small Γ there are two stagnation points on the cylinder and find their
In §29.4.1 it is asserted that eq. (29.28)describes the steady, circulation-free flow of an incompressible ideal fluid around a cylinder. To confirm this, verify (a) that va → v∞ x̂ as r
Suppose that in a region of steady flow the velocity of a fluid is given by v = Ω × r, where r is the vector from a fixed point in space. Describe the streamlines of this flow. Show that ∇
Show that the flow v around a cylinder given in eq. (29.18) has the same functional form as the magnetic field B around a long, straight wire (3.50) and therefore Maxwell?s equations imply that ? ?
A large wind turbine blade is 50 m long and has a maximum chord length of 4 m at a point one-third of the way out from the wind turbine hub. When rotating at its design rate, the blade sees an air
The dependence of Reynold?s number on the parameters of fluid flow can be inferred from dimensional analysis. First show that it is not possible to construct a dimensionless number out of the
Show that for the flow of a viscous fluid through a pipe, as analyzed in Box 29.1, the energy per unit mass dissipated as thermal energy by viscous forces is dQ/dm = Δp/ρ, where Δp is the pressure
The maximum takeoff mass for a Boeing 777-300ER passenger jet is 3.5 × 105 kg. The plane’s wing area is about 440 m2. Compute the average air pressure on the plane’s wings required to overcome
Estimate the speed of water emerging from an opening at the bottom of a dam when the water in the reservoir behind the dam is held at a height h above the opening. Compare the results of using
Supply the missing steps in the derivation of the expression for the flow rate Q in a Venturi flow meter in Example 29.2.
A stream of water emitted by a tap at velocity v0 as in Figure 29.3 has a circular cross section with radius r0. Use conservation of mass to predict how its radius changes as it falls. (a) (b)
An entrepreneur claims to have devised a system for extracting thermal energy from the wind by cooling it. His device lowers the temperature of the air that passes through it by 1 ◦C. He claims to
Angel Falls in Venezuela, at a height of 979 m, is the highest waterfall in the world. Ignoring air resistance, how much warmer is the water at the bottom of the falls, where the kinetic energy it
The forces on a parcel of moving air in the Ekman layer include the pressure force f p = ??p, the Coriolis force f hc = f v ? n? (where f = 2 ? sin ?), and a frictional force directed opposite to the
Assume that the pressure in a volume of air can be expressed to first order in a local coordinate system (x, y, z) as p = p0 + f x + O(x2, y2, xy, . . .). Show from the balance of forces on a small
The following wind frequency distribution data were obtained in the month of January at a height of 10 m above an airport runway in Buffalo, New York. Calm: 2.0%; 0–5 kt: 5.3%; 5–12 kt: 40.1%;
Use the data given in the wind rose Figure 28.19 To make a crude estimate of the average ?v? and root mean cube v? wind speed at Klamath Lake, Oregon. From these, estimate the Weibull parameters
Reference [192] studies wind power prospects for several locations in Saudi Arabia, among them Yenbu (Y) and Al Qaysumah (A). They report Weibull parameters measured at a height of 10 m: λY ≅ 5.9
Consider building a wind turbine at a wind power class 5 site. Assume that the hub height is 50 m and the blade length is 15 m, and that the turbine operates at 50% of the maximum theoretical
A steady wind is blowing at 8 m/s at 200 m above ground level. Estimate the wind power at a height of 10 m above a large open cropped field. Approximately what height is needed for a turbine in a
Suppose the wind speed obeys a Gaussian frequency distribution in the two components of the horizontal velocity v = (v1, v2), Show that the wind speed distribution, f (v) = dP/dv, Where? is a
What is the maximum power that could be extracted by a mechanical device of cross-sectional area 10 m2 in a steady wind at 10 m/s?
Compute the energy density and power density in a steady wind at 8 m/s and at 16 m/s.
The wind speed at any site can be written as the sum of its average 〈v〉 plus a fluctuating term that averages to zero, v(t) = 〈v〉 + δv(t). Show that the effect of the fluctuations is always
Suppose a gradient wind (see Box 28.1) is flowing along a circular isobar with radius R at a latitude of 45◦. Compare the strength of the Coriolis and centrifugal forces on this flow as a function
Evaluate the Coriolis factor f = 2Ωρ sin λ at ±45◦ latitude. How large a pressure gradient (in millibars/km) is necessary to sustain a geostrophic wind at 30 m/s at this latitude?
Derive the relation du/dt = d∗u/dt + ω × u by an explicit computation in a coordinate system where ω = ωẑ as described in Box 27.1.
Assume that a constant wind at v = 20 knots (10.3 m/s) is blowing in a southward direction on the eastern side of an ocean basin at latitude +35◦. The combination of the force due to the wind and
Compare the gravitational force and horizontal Coriolis force on a person running at 6 m/s at latitude 45◦. How fast would an object have to be moving for the two to be equal?
Given Earth’s radius, mass, and rate of rotation, estimate the difference in radius measured along the equator from that measured to a pole in order that the apparent gravitational potential is the
Under the same assumptions as for Problem 27.3, estimate the net rate at which energy is lost to upward thermal radiation. First, compute the rate of upward thermal radiation. Then include a downward
Consider a location in the tropical ocean where the mixed layer has a uniform temperature of 30 ◦C to 100 m, and the deep ocean temperature is 5 ◦C at 1000 m. Estimate the rate of conductive heat
Crop residues (corn stover, straw, or sugarcane bagasse) represent a potentially sustainable source of cellulosic ethanol. Assuming a production of 6 t/ha y and 270 liters of ethanol per tonne of
Compute the rate at which carbon is extracted from the atmosphere by a forest growing at the net rate of 4 t/ha y. Assume that the wood is purely composed of cellulose. Compare the carbon savings
Compute the maximum energy content of the ethanol produced from corn farmed at 10 t/ha y, assuming that the corn is 70% starch composed of polysaccharides (amylose and amylopectin) built from glucose
Look up the standard enthalpies of formation of glucose and ethanol (Table 9.8), and verify that eq. (26.4)is exothermic. Then verify that the gravimetric energy density (relative to combustion) of
If all land area in the US that is used for growing corn for ethanol were used for solar thermal electric plants operating at a gross conversion efficiency of 3%, estimate the total electrical energy
The fossil fuel energy input currently needed for ammonia production is roughly 36 GJ/t (§33.3.4). Estimate the energy needed to produce ammonia containing 90 Mt of nitrogen. Compare to the biomass
A feedlot cow is fed 12 kg of corn daily for 255 days, and grows from 200 kg to 500 kg in that time. Estimating the energy densities of corn and cow both at roughly 15 MJ/kg (a very rough estimate,
Estimate the total food energy needed for the planet’s population assuming a diet of 2400 Calories/day/person. Compare to the global food production rate stated in the text.
The enthalpy of combustion of glucose (C6H12O6) is roughly 15.6 MJ/kg. Compute the fraction of incident solar energy from 8 photons with wavelength 680 nm stored through the reaction (26.2)in a
The thickness of the germanium (density 5.32 gm/c m 3) layer in a triple-junction PV is typically greater than 100 microns. If the overall cell efficiency is 40%, how many watts of PV capacity
Assuming a silicon P V array attains 20% efficiency over a lifetime of 25 years of use, where it is exposed to an average of 250 W/ m2, what is the total energy output of 1 m2 of PV cells (in
Use the photo diode equation (25.17)to compute Iphoto/I0 for a silicon solar cell with Voc = 0.7 V. Write the power as a function of voltage and compute the fill factor, the maximum power IV
The direct band gap in silicon is 3.4 eV. What is the maximum possible collection efficiency (for incident thermal radiation at 6000 K) for excitations over this gap?
Estimate the maximum possible collection efficiency ηmaxcollection for a germanium solar cell (Egap ≅ 0.66 eV).
Determine the maximum concentration C of a solar concentrator satisfying σT4 = CI0, where I0 is the solar constant, directly from the second law of thermodynamics and the surface temperature of
For the power tower geometry from the previous problem, assume that the absorber height is equal to its diameter, and that the diameter is that found in part (b) above. If all the light from an
Consider a solar thermal “power tower” with a central tower of height 60 m, surrounded by an array of planar mirrors on the ground extending to a radius of 100 m around the tower. At the top of
Consider an idealized flat-plate solar collector with no conduction or convection losses, and with insolation 1000W/m2 incident at an angle of 45◦ from the vertical. Assume that the collector is
Consider the examples of single- and double-pane glass-covered collectors described in §24.2.1. Verify the quoted results for the efficiency as a function of Tb, Tg, and I0. In the case of double
Consider a flat-plate collector covered by a single pane of glass exposed to perpendicular insolation at 1000 W/m2. The collector operates at a net efficiency of 35% at a temperature Tb ≅ 65 ◦C.
Analyze the double pane glass-covered collector described in §24.1.2 and illustrated in Figure 24.14. Tg is assumed to be fixed at ≈ 300 K by conduction and convection and I0 ≈ 1000 W/m2. Write
Consider an idealized flat-plate solar collector. Assume that there are no heat losses to conduction or convection. Assume that the insolation is 1000 W/m2 incident at an angle of 45◦ from the
Suppose that the surface temperature of the Sun were to increase by 3% from the current value, assuming for simplicity that the radius of the Sun stays fixed. By how much would the solar constant on
The population of the city of Cambridge in the state of Massachusetts is around 100 000. Assume that each resident of Cambridge uses energy at the rate of 1 GJ per day. If the average insolation in
Suppose a gas has an absorption cross section per molecule σ(ω) for light of frequency ω (see §18.2 for an introduction to cross sections). Show that the gas’s absorption coefficient is given
For your location, calculate the maximum instantaneous irradiance (at the top of the atmosphere) on the days of the summer and winter solstices. Now suppose Earth’s obliquity were 0◦. How large
Consider a location for a solar energy installation in Arizona, at latitude 34◦. Compute the declination δon the following dates: (i) March 20 (spring equinox),(ii) April 20. Compute the length of
Show that the insolation averaged over the year and over the entire Earth is ≅ 341 W/m2.
The average radius of Mars’ orbit is 2.28 × 108 km. Compute the solar constant for Mars.
In ?7, we showed that the energy of a particle of mass M in an L ? L ? L box in the state labeled |n, m, l? is E n, m, l = ?2?2(n2 + m2 + l2)/2ML2. The probability of finding a particle in the state
Show that the classical Rayleigh?Jeans law for radiation follows from eq. (22.13) ? in the limit as ? ? 0.. Show that the total power radiated diverges in this limit (the ultraviolet catastrophe
Use dimensional analysis to show that the wavelength scale of blackbody radiation is given by λth = hc/kBT . The average radiation energy in a cavity depends only on its volume and not on its shape
Rewrite the black body power spectrum as a function of ? and compute the value ?max?where the power density dP/d? is maximized. Find the frequency ? corresponding to ?max. Why is ? not equal to ?max,
Show that the peak of the black body spectrum as a function of ? is given by eq. (22.14) kg T Wmax = 2.82
Show that the combination of the Boltzmann factor and the tunneling probability give a probability for fusion that is maximized at Emax ≈ (bkBTc/2)2/3, as stated in §22.1.1.
Compute the energy released in each step of the solar PPI fusion chain eq. (22.5) and confirm that the total energy released matches? eq. (22.6) l + |11 → {U + e+ + H + H → He + Y He + He +

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