Heat transfer between two fluids through a sandwiched solid (coupled problem):Two fluids&nbsp;(th,&nbsp;tc)&nbsp;flow opposite to each other on either side of a solid (T), while exchanging heat among themselves. In such a scenario, the conduction in the solid is governed by:x∈[0,1],y∈[0,1]1)&nbsp;κd2Tdx2+μbh(th-T)-νbc(T-tc)=0with boundary condition as&nbsp;T′(0)=T′(1)=0The fluids are governed by the following equations:2)&nbsp;dthdx+bh(th-T)=03)&nbsp;dtcdx+bc(T-tc)=0The hot fluid initiates at&nbsp;x=0&nbsp;and the cold fluid starts from&nbsp;x=1.The boundary conditions are&nbsp;th(x=0)=1&nbsp;and&nbsp;tc(x=1)=0Equation (1), (2) and (3) form a coupled system of ordinary differential equations.It is pretty evident that using (2) and (3), Equation (1) can be re-written as:4)&nbsp;κd2Tdx2-μdthdx+νdtcdx=0However, I have not been able to proceed further.Some parameter values arebc=12.38,&nbsp;bh=25.32,&nbsp;μ=1.143,&nbsp;ν=1,&nbsp;κ=2.16

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Heat transfer between two fluids through a sandwiched solid (coupled problem):Two fluids&amp;nbsp;(th,&amp;nbsp;tc)&amp;nbsp;flow opposite to each other on either side of a solid (T), while exchanging heat among themselves. In such a scenario, the conduction in the solid is governed by:x∈[0,1],y∈[0,1]1)&amp;nbsp;κd2Tdx2+μbh(th-T)-νbc(T-tc)=0with boundary condition as&amp;nbsp;T′(0)=T′(1)=0The fluids are governed by the following equations:2)&amp;nbsp;dthdx+bh(th-T)=03)&amp;nbsp;dtcdx+bc(T-tc)=0The hot fluid initiates at&amp;nbsp;x=0&amp;nbsp;and the cold fluid starts from&amp;nbsp;x=1.The boundary conditions are&amp;nbsp;th(x=0)=1&amp;nbsp;and&amp;nbsp;tc(x=1)=0Equation (1), (2) and (3) form a coupled system of ordinary differential equations.It is pretty evident that using (2) and (3), Equation (1) can be re-written as:4)&amp;nbsp;κd2Tdx2-μdthdx+νdtcdx=0However, I have not been able to proceed further.Some parameter values arebc=12.38,&amp;nbsp;bh=25.32,&amp;nbsp;μ=1.143,&amp;nbsp;ν=1,&amp;nbsp;κ=2.16

microsgopx6z7 · Accepted Answer

Step 1Hint.Calling T1=T and T2=T1′ we have(T1′T2′th′tc′)=(0100μbhκ+νbcκ0−μbhκ−νbcκbh0−bh0−bc00bc)(T1T2thtc)

Heat transfer between two fluids through a sandwiched

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