Advertisements
Advertisements
प्रश्न
Find the rate of heat flow through a cross section of the rod shown in figure (28-E10) (θ2 > θ1). Thermal conductivity of the material of the rod is K.
Advertisements
उत्तर
We can infer from the diagram that ΔPQR is similar to ΔPST.
So, by the property of similar triangles,
`x/l = (r - r_1)/(r_2 - r_1)`
⇒ `(r_2 - r_1) x/l = r - r_1`
⇒ `r = r_1 + ( r_2 - r_1 ) x/l`
Let ;
`(r_2 - r_1)/l = a`
⇒ r = ax + r_1................(i)
Thermal resistance is given by
`dR = dx/(K.A)`
⇒ dR = `(dx)/(K.pir^2`]
`dR = (dx)/(K.pi (ax + r_1)^2`
\[\int\limits_0^R\] dR = `1/(kpi)` \[\int\limits_0^l\] `dx/(ax + r_1)`
`R = -(1)/(Kpi) [(1)/(ax + r_1)]_0^6`
`R = -(1)/(aKpi)[(1)/(al + r_1) -1/(r1)]`
`R = -(1)/(aKpi) [ 1/(((r_2 -r_1)/l) l + r_1) - 1/r_1]`
`R = - 1/ (aKpi)[(1)/(r_2)-(1)/(r_1)]`
`R = -(1)/(Kpir_1r_2)`
Rate of flow of heat = `(DeltaQ)/R`
`⇒ q = `( Q_2 - Q_1)/l Kpir_1r_2`
APPEARS IN
संबंधित प्रश्न
A bullet of mass 20 g enters into a fixed wooden block with a speed of 40 m s−1 and stops in it. Find the change in internal energy during the process.
A brick weighing 4.0 kg is dropped into a 1.0 m deep river from a height of 2.0 m. Assuming that 80% of the gravitational potential energy is finally converted into thermal energy, find this thermal energy is calorie.
The blocks of masses 10 kg and 20 kg moving at speeds of 10 m s−1 and 20 m s−1respectively in opposite directions, approach each other and collide. If the collision is completely inelastic, find the thermal energy developed in the process.
The thermal conductivity of a rod depends on
One end of a metal rod is kept in a furnace. In steady state, the temperature of the rod
A hot liquid is kept in a big room. The logarithm of the numerical value of the temperature difference between the liquid and the room is plotted against time. The plot will be very nearly
A piece of charcoal and a piece of shining steel of the same surface area are kept for a long time in an open lawn in bright sun.
(a) The steel will absorb more heat than the charcoal
(b) The temperature of the steel will be higher than that of the charcoal
(c) If both are picked up by bare hand, the steel will be felt hotter than the charcoal
(d) If the two are picked up from the lawn and kept in a cold chamber, the charcoal will lose heat at a faster rate than the steel.
A liquid-nitrogen container is made of a 1 cm thick styrofoam sheet having thermal conductivity 0.025 J s−1 m−1 °C−1. Liquid nitrogen at 80 K is kept in it. A total area of 0.80 m2 is in contact with the liquid nitrogen. The atmospheric temperature us 300 K. Calculate the rate of heat flow from the atmosphere to the liquid nitrogen.
Water at 50°C is filled in a closed cylindrical vessel of height 10 cm and cross sectional area 10 cm2. The walls of the vessel are adiabatic but the flat parts are made of 1-mm thick aluminium (K = 200 J s−1 m−1°C−1). Assume that the outside temperature is 20°C. The density of water is 100 kg m−3, and the specific heat capacity of water = 4200 J k−1g °C−1. Estimate the time taken for the temperature of fall by 1.0 °C. Make any simplifying assumptions you need but specify them.
The ends of a metre stick are maintained at 100°C and 0°C. One end of a rod is maintained at 25°C. Where should its other end be touched on the metre stick so that there is no heat current in the rod in steady state?
A metal rod of cross sectional area 1.0 cm2 is being heated at one end. At one time, the temperatures gradient is 5.0°C cm−1 at cross section A and is 2.5°C cm−1 at cross section B. Calculate the rate at which the temperature is increasing in the part AB of the rod. The heat capacity of the part AB = 0.40 J°C−1, thermal conductivity of the material of the rod = 200 W m−1°C−1. Neglect any loss of heat to the atmosphere
A hollow tube has a length l, inner radius R1 and outer radius R2. The material has a thermal conductivity K. Find the heat flowing through the walls of the tube if (a) the flat ends are maintained at temperature T1 and T2 (T2 > T1) (b) the inside of the tube is maintained at temperature T1 and the outside is maintained at T2.
An aluminium rod and a copper rod of equal length 1.0 m and cross-sectional area 1 cm2 are welded together as shown in the figure . One end is kept at a temperature of 20°C and the other at 60°C. Calculate the amount of heat taken out per second from the hot end. Thermal conductivity of aluminium = 200 W m−1°C−1 and of copper = 390 W m−1°C−1.
Following Figure shows an aluminium rod joined to a copper rod. Each of the rods has a length of 20 cm and area of cross section 0.20 cm2. The junction is maintained at a constant temperature 40°C and the two ends are maintained at 80°C. Calculate the amount of heat taken out from the cold junction in one minute after the steady state is reached. The conductivites are KAt = 200 W m−1°C−1 and KCu = 400 W m−1°C−1.
The three rods shown in figure have identical geometrical dimensions. Heat flows from the hot end at a rate of 40 W in the arrangement (a). Find the rates of heat flow when the rods are joined as in arrangement (b) and in (c). Thermal condcutivities of aluminium and copper are 200 W m−1°C−1 and 400 W m−1°C−1 respectively.
A hollow metallic sphere of radius 20 cm surrounds a concentric metallic sphere of radius 5 cm. The space between the two spheres is filled with a nonmetallic material. The inner and outer spheres are maintained at 50°C and 10°C respectively and it is found that 100 J of heat passes from the inner sphere to the outer sphere per second. Find the thermal conductivity of the material between the spheres.
A spherical ball of surface area 20 cm2 absorbs any radiation that falls on it. It is suspended in a closed box maintained at 57°C. (a) Find the amount of radiation falling on the ball per second. (b) Find the net rate of heat flow to or from the ball at an instant when its temperature is 200°C. Stefan constant = 6.0 × 10−8 W m−2 K−4.
