A heat exchanger is made up of two identical pipes separated by an impermeable diathermal wall of section area A, thickness h and thermal conductivity κ. In both pipes, a liquid flows at uniform velocities v₁ = v₁ ˆx and v₂ = −v₂ ˆx, with v₁ > 0 and v₂ > 0, where ˆx is the unit vector that is parallel to the liquid flow in pipe 1. The temperature T₁ of the liquid in pipe 1 is larger than the temperature T₂ of the liquid in pipe 2, i.e. T₁ > T₂. Thus, there is a heat current density j_Q = j_Qˆy, with jQ > 0 going across the wall separating the pipes, where ˆy is the unit vector orthogonal to the wall and oriented positively from pipe 1 to pipe 2. There is no liquid current density across the wall, i.e. j_C = 0. Heat conductivity is considered negligible in the direction of the flow and yet large enough to ensure a homogeneous temperature across any section of both pipes. Consider that the heat exchanger has reached a stationary state.a) Show that the temperature profiles in the fluids are given by the differential equations,where c₁ and c₂ are the specific heat densities of liquids 1 and 2, and κ is the thermal conductivity of the diathermal wall and is a characteristic length for the thermal transfer.b) Show that the convective heat current density j = c₁ v₁ T₁ + c₂ v₂ T₂ is homogeneous.c) Determine the temperature difference ΔT (x) = T₁ (x) − T₂ (x).

d) Determine the temperature profiles T₁ (x) and T₂ (x).e) Show that on a distance that is short enough, i.e. x/d 1,

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8, T1 Əx T₂ = K hl cj vi K hlcz vz (T₁-T₂) (T₁ T₂) -