Question

A heat-transfer experiment in CBE 154 investigates conductive heat transport within a solid and convective heat transport at a solid gas interface. In this homework problem, you are to write the differential equations describing the experimental setup (but not solve them). The first part of the experiment involves measurement of a steady-state temperature profile along a brass rod to determine the conductivity of brass and the rate of heat loss through the insulation to the surrounding air. The second part characterizes heat transfer from a hot brass face to a jet of cooling air blown perpendicularly to the opposing face (called “impinging jet flow"). In CBE 154, you are to determine the heat- transfer coefficient for the air-cooled brass surface and compare this value to that obtained from correlations. Figure 53 shows a schematic of the experimental apparatus. The cylindrical brass rod is resistively heated at the right face with the face temperature measured by a thermocouple (not shown). The left end of the brass rod is cooled by air flow of known temperature and flow rate. The brass rod is insulated by solid nylon. However, insulation is not complete. There is some heat loss through the insulation. You are to write the differential equations describing the heat transfer in the apparatus (brass rod and insulation). Boundary conditions are especially important. In CBE 154, you will numerically solve this problem (see what you have to look forward to). Here, however, you are only to list the governing equations and boundary conditions.
please show all steps!

Answer 1

1) select control volume of thickness delta z and delta r radius

2) do heat balance and form differential equation

3) boundary condition

1. 1st BC :  Temperature is know at 1 end using thermocouple and constant hence T(r,z) = T(r,L) = Thermocouple temperature 2
2. 2nd BC : flux at z = 0 is equal to rate of convective heat transfer -> d( T(r,0) /dz = - h A (T - Tair) ( Tair = bulk air fluid temperature, h = heat transfer coefficient, T = temperature of brass at Z =0 )

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