Advertisements
Advertisements
प्रश्न
Two bodies of masses m1 and m2 and specific heat capacities s1 and s2 are connected by a rod of length l, cross-sectional area A, thermal conductivity K and negligible heat capacity. The whole system is thermally insulated. At time t = 0, the temperature of the first body is T1 and the temperature of the second body is T2 (T2 > T1). Find the temperature difference between the two bodies at time t.
Advertisements
उत्तर
Rate of transfer of heat from the rod is given by
`(DeltaQ)/(Deltat) = (KA(T_2 - T_1))/l`
Heat transfer from the rod in time ΔΔ t is given by
`(DeltaQ)/(Deltat) = (KA(T_2 - T_1))/l Deltat ............(1)`
Heat loss by the body at temperature T2 is equal to the heat gain by the body at temperature T1
Therefore, heat loss by the body at temperature t2 in time Δt is given by
`DeltaQ = m_2s_2(T_2 - T_2) ....(2)`
from equation (i) and (ii)
`m_2s_2(T_2 - T_2')= (KA(T_2 - T_1))/l Delta t`
`⇒ T_2' = T_2 - (KA(T_2 - T_1))/(l(m_2s_2)) Delta t`
This gives us the fall in the temperature of the body at temperature T2.
Similarly, rise in temperature of water at temperature T1 is given by
`T_1' = T_1 + (KA(T_2 - T_1))/(l(m_1s_1)) Delta t`
Change in the temperature is given by
`(T_2' - T_1') = (T_2 - T_1) - [(KA (T_2 - T_1))/(lm_1s_1) Deltat + (KA(T_2 - T_1))/(lm_2s_1)Delta t]`
`⇒(T_2' - T_1') - (T_2 - T_1) = - [(KA(T_2 - T
_1))/(lm_1s_1) Deltat + [(KA(T_2 - T
_1))/(lm_2s_2) Deltat]`
`rArr (DeltaT)/(Deltat)= (KA(T_2 - T_1))/l [1/(m_1s_1) + 1/(m_2 s_2)] Deltat`
`rArr 1/(T_2 - T_1) DeltaT =- (KA)/l [(m_1s_1 + m_2s_2)/(m_1s_1m_2s_2)] `
On integrating both the sides, we get
lim Δ t → 0
`int 1/(T_2 - T_1)dT = int - (KA)/l [( m_1s_1 + m_2s_2)/(m_1s_1m_2s_2) ]dt`
⇒ `In [T_2 - T_1] = - (KA)/l [( m_1s_1 + m_2s_2)/(m_1s_1m_2s_2)]t`
⇒ `(T_2 - T_1) = e^(-lamda t)`
Here , `lamda = "KA/l [ "m_1s_1 + m_2s_2"/"m_1s_1m_2s_2"]`
APPEARS IN
संबंधित प्रश्न
A van of mass 1500 kg travelling at a speed of 54 km h−1 is stopped in 10 s. Assuming that all the mechanical energy lost appears as thermal energy in the brake mechanism, find the average rate of production of thermal energy is cal s−1.
A block of mass 100 g slides on a rough horizontal surface. If the speed of the block decreases from 10 m s−1 to 5 m s−1, 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 uniform slab of dimension 10 cm × 10 cm × 1 cm is kept between two heat reservoirs at temperatures 10°C and 90°C. The larger surface areas touch the reservoirs. The thermal conductivity of the material is 0.80 W m−1 °C−1. Find the amount of heat flowing through the slab per minute.
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 is boiled in a container having a bottom of surface area 25 cm2, thickness 1.0 mm and thermal conductivity 50 W m−1°C−1. 100 g of water is converted into steam per minute in the steady state after the boiling starts. Assuming that no heat is lost to the atmosphere, calculate the temperature of the lower surface of the bottom. Latent heat of vaporisation of water = 2.26 × 106 J kg−1.
A icebox almost completely filled with ice at 0°C is dipped into a large volume of water at 20°C. The box has walls of surface area 2400 cm2, thickness 2.0 mm and thermal conductivity 0.06 W m−1°C−1. Calculate the rate at which the ice melts in the box. Latent heat of fusion of ice = 3.4 × 105 J kg−1.
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.
The two rods shown in following figure have identical geometrical dimensions. They are in contact with two heat baths at temperatures 100°C and 0°C. The temperature of the junction is 70°C. Find the temperature of the junction if the rods are interchanged.
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.
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.
