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Mechanical Properties Of Fluids

Question
CBSEENPH11019794

Two slabs A and B of thermal conductivity Kl and K2, length l2 and l2 and each of cross section area A are placed in scries. The free end of slab A is at temperature T1 and that of slab B is T2 <T1. Find the temperature at the interface of two slabs. Also find the rate of flow of heat through compound slab in steady state.

Solution

Let T be the temperature at the interface of two slabs
                    
The rate of flow of heat through slab A is,
dQ subscript 1 over dt equals fraction numerator straight K subscript 1 straight A over denominator straight l subscript 1 end fraction left parenthesis straight T subscript 1 minus straight T right parenthesis space space space space space space space space space space space... space left parenthesis 1 right parenthesis
The rate of flow of heat through slab B is, 

dQ subscript 2 over dt equals fraction numerator straight K subscript 2 straight A over denominator straight l subscript 1 end fraction left parenthesis straight T minus straight T subscript 2 right parenthesis space space space space space space space space space space space space space... space left parenthesis 2 right parenthesis
As two slabs are in series, therefore the rate of flow of heat through them will be same.
straight i. straight e. space space space space space space space space space space space space space dQ over dt equals dQ subscript 2 over dt space

space space space space space space space straight K subscript 1 over straight l subscript 1 left parenthesis straight T subscript 1 minus straight T right parenthesis equals straight K subscript 2 over straight l subscript 2 left parenthesis straight T minus straight T subscript 2 right parenthesis space

space straight K subscript 1 over straight l subscript 1 straight T subscript 1 plus straight K subscript 2 over straight l subscript 2 straight T subscript 2 equals straight K subscript 1 over straight l subscript 1 straight T plus straight K subscript 2 over straight l subscript 2 straight T space

rightwards double arrow space space space space straight T equals fraction numerator space space space space space space space space space space begin display style straight K subscript 1 over straight l subscript 1 end style straight T subscript 1 plus begin display style straight K subscript 2 over straight l subscript 2 end style straight T subscript 2 space over denominator begin display style straight K subscript 1 over straight l subscript 1 plus straight K subscript 2 over straight l subscript 2 end style end fraction space

space space space space space space space space space space equals fraction numerator straight l subscript 2 straight K subscript 1 straight T subscript 1 plus straight l subscript 1 straight K subscript 2 straight T subscript 2 over denominator straight l subscript 2 straight K subscript 1 plus straight l subscript 1 straight K subscript 2 end fraction
The rate of flow of heat through composite slab is,
dQ over dt equals dQ subscript 1 over dt equals fraction numerator straight K subscript 1 straight A over denominator straight l subscript 1 end fraction left parenthesis straight T subscript 1 minus straight T right parenthesis

space space space space space space space space equals fraction numerator straight K subscript 1 straight A over denominator straight l subscript 1 end fraction open parentheses straight T subscript 1 minus fraction numerator straight l subscript 2 straight K subscript 1 straight T subscript 1 plus straight l subscript 1 straight K subscript 2 straight T subscript 2 over denominator straight l subscript 2 straight K subscript 1 plus straight l subscript 1 straight K subscript 2 end fraction close parentheses space

space space space space space space space space equals fraction numerator straight K subscript 1 straight K subscript 2 over denominator straight l subscript 2 straight K subscript 1 plus straight l subscript 1 straight K subscript 2 end fraction straight A left parenthesis straight T subscript 1 minus straight T subscript 2 right parenthesis