2.55   Consider a one-dimensional plane wall of thickness 2L. The   surface   at   x = -L   is   subjected   to   convective conditions characterized by T00,1, h1, while the surface

 

 

 

at x = +L is subjected to conditions T00,2, h2. The initial temperature of the wall is To = (T00,1 + T00,2)/2 where T00,1 > T00,2.

 

 

 

 

 

To h1, TÂ¥,1

 

TÂ¥,1 > TÂ¥,2

 

 

 

2L

 

 

 

 

 

 

 

x

 

 

 

 

 

(a)    Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature distribution T(x, t) as a function of position and time.

 

(b)    On T - x coordinates, sketch the temperature dis- tributions for the initial condition, the steady-state condition, and for two intermediate times for the case h1 = h2.

 

(c)    On q"x  - t coordinates, sketch the heat flux qx"(x, t) at the planes x = 0, -L, and +L.

 

(d)    The value of h1 is now doubled with all other con- ditions being identical as in parts (a) through (c). On T - x coordinates drawn to the same scale as used in part (b), sketch the temperature distribu- tions for the initial condition, the steady-state con-

 

(c)    If h = 100 W/m2 · K, what is the heat flux at x = L and t = 0? If the wall has a thermal conductivity of k = 50 W/m · K what is the corresponding tempera- ture gradient at x = L?

 

(d)   Consider a plate of thickness 2L = 20 mm with a density of p = 2770 kg/m3  and  a  specific heat cp = 875 J/kg · K. By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate (J/m2) that is trans- ferred to the bath over the time required to reach steady-state conditions.

 

(e)    From other considerations, it is known that, during the quenching process, the heat flux at x = +L and x = -L decays exponentially with time according to the relation, qx" = A exp(-Bt), where t is in seconds, A = 1.80 X 104  W/m2, and B = 4.126 X 10-3  s-1.

 

Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process.