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 conductivities of alumimium and copper are 200 W m–1 °C–1 and 400 W m–1 °C–1 respectively.



(a).Let’s redraw the diagram


w10.png


Since all rods are connected in series,


Req = RAl + RCu + RAl


Given -


Temperature of the hot end ,T1 = 100°C


cold end ,T2 = 0°C


Substitution for Req –


Req =


Given Thermal conductivities of aluminium =200 W m–1 °C–1


Copper = 400 W m–1 °C–1


Therefore


Req = + + ) =


Now rate of flow of heat,


=


= (1)


Given rate of flow of heat


From (1)


40 = 80 × 100 ×


⇒ (2)


(b) Lets redraw the diagram –


w11.png


The equivalent circuit in terms of thermal resistances becomes-


w12.PNG


Req = RAl +


= ++


=


=


Now rate of flow of heat,


=


= (3)


From (2)


Substituting in (3)


=


= 75 W


(c) Lets redraw the diagram-


w13.PNG


Since the thermal resistors are connected in parallel.


= + +


⇒ =


⇒ = (Al + Al +Cu ) = (200+200+400)


⇒ = *(800)


⇒ =


From (2)


⇒ =


⇒ = 400W


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