(a) The inside of a hollow cylinder is maintained at a temperature Ta, and the outside is at a lower temperature, Tb (Fig. P19.45). The wall of the cylinder has a thermal conductivity k. Ignoring end effects, show that the rate of energy
Suggestions: The temperature gradient is dT/dr. A radial energy current passes through a concentric cylinder of area 2πrL. (b) The passenger section of a jet airliner is in the shape of a cylindrical tube with a length of 35.0 m and an inner radius of 2.50 m. Its walls are lined with an insulating material 6.00 cm in thickness and having a thermal conductivity of 4.00 × 10−5 cal/s · cm · °C. A heater must maintain the interior temperature at 25.0°C while the outside temperature is −35.0°C. What power must be supplied to the heater?
Figure P19.45
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Chapter 19 Solutions
Physics:f/sci.+engrs.,ap Ed.
- An aluminum rod 0.500 m in length and with a cross sectional area of 2.50 cm2 is inserted into a thermally insulated vessel containing liquid helium at 4.20 K. The rod is initially at 3(H) K. (a) If one-halt of the rod is inserted into the helium, how many liters of helium boil off by the time the inserted half cools to 4.20 K? Assume the upper half does not yet cool, (b) If the circular surface of the upper end of the rod is maintained at 300 K. what is the approximate boil-off rate of liquid helium in liters per second after the lower half has reached 4.20 K? (Aluminum has thermal conductivity of 3 100 YV/m K at 4.20 K; ignore its temperature variation. The density of liquid helium is 125 kg/m3.)arrow_forwardHow many moles are there in (a) 0.0500 g of N2 gas (M = 28.0 g/mol)? (b) 10.0 g of CO2 gas (M = 44.0 g/mol)? (c) How many molecules are present in each case?arrow_forwardConsider an object with any one of the shapes displayed in Table 10.2. What is the percentage increase in the moment of inertia of the object when it is warmed from 0C to 100C if it is composed of (a) copper or (b) aluminum? Assume the average linear expansion coefficients shown in Table 16.1 do not vary between 0C and 100C. (c) Why are the answers for parts (a) and (b) the same for all the shapes?arrow_forward
- When builders were constructing a sidewalk they forgot to include an expansion joint between two of the segments, L = 2.1 m at To = 20° C. Assume the opposite ends of each segment are fixed and the linear expansion coefficient is α = 11 × 10-6 °C-1. a.) As the day heats to Tb the segments press against each other and begin to raise the junction a distance h forming a triangle. What is the height (in meters) at Tb=110° F? b.) What should the gap have been, in units of meters, to prevent them from touching?arrow_forwardIn a 60 mm diameter cylindrical nuclear reactor fuel rod ė ü, 1 = 3.929 × 107 W / m3 uniformly distributed There is heat generation and the temperature distribution in continuous regime is given by the equation T = a + br2. Here, A = 650 ° C and b = -3.274 x 105 ° C in T (° C) and r (m) units. The properties of the rod k = 30 W / m. K, p = 1100 kg / m3 and cp = 800 J/ kg · K: (a) What is the heat conduction (W 7 m) from the unit length of the rod at r = 0 (axis) and r = mm (surface)? (b) Reactor power level suddenly ė lf ü is increased to 2 = 108 W / m3, at r = 0 andr= be What is the change (@T / at) with time? 30 mm the temperature willarrow_forwardWhen builders were constructing a sidewalk they forgot to include an expansion joint between two of the segments, L = 2.5 m at To = 20° C. Assume the opposite ends of each segment are fixed and the linear expansion coefficient is α = 11.9 × 10-6 °C-1. a) As the day heats to Tb the segments press against each other and begin to raise the junction a distance h forming a triangle. What is the height (in meters) at Tb=110° F? b) What should the gap have been to prevent them from touching in meters?arrow_forward
- An isolated bar is kept at zero temperature at its ends. If it's given initially temperature, f(x) = 100°C, a2y equation u (x, t) is equal to: a2y so the bar temperature ax2 %3D u (0, t) u (L, t) [sin#x•e) , sin x -400 -400 (sinx-e ) + TC .x. (25x² sinx e sin x-e ) The Answer Is the Answer is O Isinx e e sin 400 None of all 3. 25m sin x earrow_forwardb) The heat capacity of solid lead oxide [PbO (s)] is given by Cp.m JK- mol = 44.35 – 1.47 × 10-32 Calculate the change in molar enthalpy (AHm) (in unit J mol*) of PbO (s) if it is cooled from 500 K to 300 K at constant pressure (P). Quantity Cp,m denotes molar heat capacity at constant pressure.arrow_forwardI.C 02/A/ Use the Crank-Nicolson method to solve for the temperature distribution of a long thin rod with a length of 10 cm and the following values: k = 0.49 cal/(s cm °C), Ax = 2 cm, and At = st 0.1 s. Initially the temperature of the rod is 0°C and the boundary conditions are fixed for all times at 7(0, t) = 100°C and 7(10, t) = 50°C. Note that the rod is aluminum with C = 0.2174 cal/g °C) and p = 2.7 g/cm³. List the tridiagonal system of equations and determined the temperature up to 0.1 s.arrow_forward
- A space probe is far away from the Sun, or any other sources of energy. It is kept warm (so the electronics work) by waste heat from a radioactive source. The radioactive source emits energy at a rate of 625W. The space probe can be modelled as a uniform sphere of metal. The radius is 1.3m, and the metal has a high thermal conductivity, so the probe is at a uniform temperature. The coefficient of linear expansion for this metal is 2.3 × 10-5 K-¹. 8. Suppose that the probe emits as a blackbody (with e = 1). What is the equilibrium temperature of the space probe? (a) 30K (b) 90K (c) ***150K (d) 240K (e) 320K 9. Suppose that the space probe is currently at temperature 200K, and it was launched from Earth at a temperature of 300K. By how much has the probe's volume decreased relative to the size at launch? (a) 2.1 x 10-2m³ (b) 4.2 × 10-²m³ (c) ***6.3 × 10-²m³ (d) 8.5 × 10-2m³ (e) There is not enough information to determine this.arrow_forwardA bimetallic strip of length L is made of two ribbons of different metals bonded together. (a) First assume the strip is originally straight. As the strip is warmed, the metal with the greater average coefficient of expansion expands more than the other, forcing the strip into an arc, with the outer radius having a greater circumference (Fig. P10.61). Derive an expression for the angle of bending, u, as a function of the initial length of the strips, their average coefficients of linear expansion, the change in temperature, and the separation of the centers of the strips (Δr = r2 −r1). (b) Show that the angle of bending goes to zero when Δ T goes to zero and also when the two average coefficients of expansion become equal. (c) What happens if the strip is cooled?arrow_forwardA steam pipe is covered with 1.50-cm thick insulating material of thermal conductivity of 0.200 cal/cm · °C · s. How much energy is lost every second when the steam is at 250°C and the surrounding air is at 20.0°C? The pipe has a circumference of 800 cm and a length of 64.0 m. Neglect losses through the ends of the pipe. ?MWarrow_forward
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