A composite metal bar of uniform section is made up of length 25 cm of copper, 10 cm of nickel

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10-2.Transmission of Heat

hard

A composite metal bar of uniform section is made up of length $25 cm$ of copper, $10 cm$ of nickel and $15 cm$ of aluminium. Each part being in perfect thermal contact with the adjoining part. The copper end of the composite rod is maintained at ${100^o}C$ and the aluminium end at ${0^o}C$. The whole rod is covered with belt so that there is no heat loss occurs at the sides. If ${K_{{\rm{Cu}}}} = 2{K_{Al}}$ and ${K_{Al}} = 3{K_{{\rm{Ni}}}}$, then what will be the temperatures of $Cu - Ni$ and $Ni - Al$ junctions respectively

${23.33^o}C$ and $A$

${83.33^o}C$ and ${20^o}C$

${50^o}C$ and ${30^o}C$

${30^o}C$ and ${50^o}C$

Solution

(b) If suppose ${K_{Ni}} = K \Rightarrow {K_{Al}} = 3K$ and ${K_{Cu}} = 6K.$
Since all metal bars are connected in series

So ${\left( {\frac{Q}{t}} \right)_{Combination}} = {\left( {\frac{Q}{t}} \right)_{Cu}} = {\left( {\frac{Q}{t}} \right)_{Al}} = {\left( {\frac{Q}{t}} \right)_{Ni}}$

and $\frac{3}{{{K_{eq}}}} = \frac{1}{{{K_{Cu}}}} + \frac{1}{{{K_{Al}}}} + \frac{1}{{{K_{Ni}}}} = \frac{1}{{6K}} + \frac{1}{{3K}} + \frac{1}{K} = \frac{9}{{6K}}$

==> ${K_{eq}} = 2K$

Hence, if ${\left( {\frac{Q}{t}} \right)_{Combination}} = {\left( {\frac{Q}{t}} \right)_{Cu}}$

==> $\frac{{{K_{eq}}\,A(100 – 0)}}{{{l_{Combination}}}} = \frac{{{K_{Cu}}A(100 – {\theta _1})}}{{{l_{Cu}}}}$

==> $\frac{{2K\,A\,(100 – 0)}}{{(25 + 10 + 15)}} = \frac{{6K\,A\,(100 – {\theta _1})}}{{25}}$

==> ${\theta _1} = 83.33^\circ C$

Similar if ${\left( {\frac{Q}{t}} \right)_{Combination}} = {\left( {\frac{Q}{t}} \right)_{Al}}$

==> $\frac{{2K\,A(100 – 0)}}{{50}} = \frac{{3K\,A({\theta _2} – 0)}}{{15}}$

==> ${\theta _2} = {20^o}C$

Standard 11

Physics

When thermal conductivity is said to be constant ?

easy

The wall with a cavity consists of two layers of brick separated by a layer of air.All three layers have the same thickness and the thermal conductivity of the brick is much greater than that of air. The left layer is at a higher temperature than the right layer and steady state condition exists. Which of the following graphs predicts correctly the variation of temperature $T$ with distance $d$ inside the cavity?

normal

Two rods $A$ and $B$ of same cross-sectional are $A$ and length $l$ connected in series between a source $(T_1 = 100^o C)$ and a sink $(T_2 = 0^o C)$ as shown in figure. The rod is laterally insulated If $G_A$ and $G_B$ are the temperature gradients across the rod $A$ and $B$, then

medium

An iron bar $\left(L_{1}=0.1\; m , A_{1}\right.$ $\left.=0.02 \;m ^{2}, K_{1}=79 \;W m ^{-1} K ^{-1}\right)$ and a brass bar $\left(L_{2}=0.1\; m , A_{2}=0.02\; m ^{2}\right.$ $K_{2}=109 \;Wm ^{-1} K ^{-1}$ are soldered end to end as shown in Figure. The free ends of the iron bar and brass bar are maintained at $373 \;K$ and $273\; K$ respectively. Obtain expressions for and hence compute

$(i)$ the temperature of the junction of the two bars,

$(ii)$ the equivalent thermal conductivity of the compound bar, and

$(iii)$ the heat current through the compound bar.

medium

$A$ metal rod of length $2$$m$ has cross sectional areas $2A$ and $A$ as shown in figure. The ends are maintained at temperatures $100°C$ and $70°C$ . The temperature at middle point $C$ is…… $^oC$

medium

Solution

Similar Questions

When thermal conductivity is said to be constant ?

Two rods $A$ and $B$ of same cross-sectional are $A$ and length $l$ connected in series between a source $(T_1 = 100^o C)$ and a sink $(T_2 = 0^o C)$ as shown in figure. The rod is laterally insulated If $G_A$ and $G_B$ are the temperature gradients across the rod $A$ and $B$, then

$A$ metal rod of length $2$$m$ has cross sectional areas $2A$ and $A$ as shown in figure. The ends are maintained at temperatures $100°C$ and $70°C$ . The temperature at middle point $C$ is…… $^oC$

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