Solder is made from various alloys designed to melt upon contact with a very hot surface...

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Solder is made from various alloys designed to melt upon contact with a very hot surface (i.e., metal piece which is to be joined to another). As shown in figure. below, assume that a solder wire of radius R is melting at a constant rate, yielding a constant wire velocity u The melting temperature of the solder is Tm. the ambient temperature is T. and the latent heat of fusion is A. The convection heat transfer coefficient h from the wire to the ambient air is assumed to be constant. The wire is long enough that it eventually reaches the ambient temperature far from the surface. 2R- b) Solder wire Air h To Tm a) Derive a differential equation to obtain temperature profile as a function of axial (z) and radial (r) directions, T(r,z). Write the boundary conditions. Propose a solution method for the differential equation driven. Explain the reasons of the method you suggest. Do not solve the differential equation. Under what conditions is a one-dimensional analysis valid for the temperature profile in the wire (le., T = T(z) only)? c) Obtain the temperature profile, T(z), for the conditions of part (b) in case of low wire velocity (i.e: conduction in the wire should be considered) d) Derive an expression for the heat loss from the wire to the environment. Hot surface Solder is made from various alloys designed to melt upon contact with a very hot surface (i.e., metal piece which is to be joined to another). As shown in figure. below, assume that a solder wire of radius R is melting at a constant rate, yielding a constant wire velocity u The melting temperature of the solder is Tm. the ambient temperature is T. and the latent heat of fusion is A. The convection heat transfer coefficient h from the wire to the ambient air is assumed to be constant. The wire is long enough that it eventually reaches the ambient temperature far from the surface. 2R- b) Solder wire Air h To Tm a) Derive a differential equation to obtain temperature profile as a function of axial (z) and radial (r) directions, T(r,z). Write the boundary conditions. Propose a solution method for the differential equation driven. Explain the reasons of the method you suggest. Do not solve the differential equation. Under what conditions is a one-dimensional analysis valid for the temperature profile in the wire (le., T = T(z) only)? c) Obtain the temperature profile, T(z), for the conditions of part (b) in case of low wire velocity (i.e: conduction in the wire should be considered) d) Derive an expression for the heat loss from the wire to the environment. Hot surface