A small bore tubular reactor with plug flow velocity u01 ms

A small bore tubular reactor with plug flow velocity u=0.1 m/s loses heat through the wall to the surroundings held at temperature T_a =300K, at a rate given by Newton\'s law of cooling with a heat transfer coefficient h=40 W/m^2K all along the tube length from z=0 to z=L. The forward reaction A rightarrow B is first order and exothermic with the heat of reaction DeltaH_R = -80 kJ/mole. The concentration and temperature at the inlet z=0 are c_A0=1 mole/m^3 and T_0=360K; pC_p=1.6 kJ/m^3K and L=2.5 m. The reaction rate is given as: r = k(T)c_A and k(T) = k(T_0) exp [-E_a/RT + E_a/RT_0] where k(T_0)=0.4 s^-1 and E_a/R=7400K. Since the tube radius is small-R=0.01m, we take the reactor temperature T to vary only along the axial coordinate z. (a) Carry out a steady-state energy balance on an appropriate shell within the tube, accounting for convection, heat loss through the wall and heat of reaction to get an ordinary differential equation for T(z). Is the temperature specification at the inlet enough to solve this ODE? What else do you need? (b) Show that when the differential equation from (a) for T(z) is scaled with the usual choices such as L for scaling z, T_0 for T and C_A0 for C_A, four dimensionless groups arise in the dimensionless ordinary differential equation as listed below. Da = k(T_0)L/u beta = (-DeltaH_R) C_A0/pC_pT_0 gamma = E_a/RT_0 delta = 2hL/RuPC_p (c) Explain the physical significance of each group - what rates or times is it a ratio of-- or and evaluate each group for the given conditions. Take care: 1 W= 1 J/s but some other parameters are given in kJ. Write the scaled differential equation with these groups in it

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