Fundamentals of heat and mass transfer frank p incropera david p dewitt solution manual ch05

PROBLEM 5.1 KNOWN: Electrical heater attached to backside of plate while front surface is exposed to convection process (T∞,h); initially plate is at a uniform temperature of the ambient air and suddenly heater power is switched on providing a constant q′′o FIND: (a) Sketch temperature distribution, T(x,t), (b) Sketch the heat flux at the outer surface, q′′x ( L,t ) as a function of time SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction, (2) Constant properties, (3) Negligible heat loss from heater through insulation ANALYSIS: (a) The temperature distributions for four time conditions including the initial distribution, T(x,0), and the steady-state distribution, T(x,∞), are as shown above Note that the temperature gradient at x = 0, -dT/dx)x=0, for t > will be a constant since the flux, q′′x ( ), is a constant Noting that To = T(0,∞), the steady-state temperature distribution will be linear such that T − T ( L,∞ ) q′′o = k o = h T ( L,∞ ) − T∞  L (b) The heat flux at the front surface, x = L, is given by q′′x ( L,t ) = −k ( dT/dx ) x=L From the temperature distribution, we can construct the heat flux-time plot COMMENTS: At early times, the temperature and heat flux at x = L will not change from their initial values Hence, we show a zero slope for q′′x ( L,t ) at early times Eventually, the value of q′′x ( L,t ) will reach the steady-state value which is q′′o PROBLEM 5.2 KNOWN: Plane wall whose inner surface is insulated and outer surface is exposed to an airstream at T∞ Initially, the wall is at a uniform temperature equal to that of the airstream Suddenly, a radiant source is switched on applying a uniform flux, q′′o , to the outer surface FIND: (a) Sketch temperature distribution on T-x coordinates for initial, steady-state, and two intermediate times, (b) Sketch heat flux at the outer surface, q′′x ( L,t ) , as a function of time SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction, (2) Constant properties, (3) No internal generation, E g = 0, (4) Surface at x = is perfectly insulated, (5) All incident radiant power is absorbed and negligible radiation exchange with surroundings ANALYSIS: (a) The temperature distributions are shown on the T-x coordinates and labeled accordingly Note these special features: (1) Gradient at x = is always zero, (2) gradient is more steep at early times and (3) for steady-state conditions, the radiant flux is equal to the convective heat flux (this follows from an energy balance on the CS at x = L), q ′′o = q ′′conv = h [T ( L,∞ ) − T∞ ] (b) The heat flux at the outer surface, q′′x ( L,t ) , as a function of time appears as shown above COMMENTS: The sketches must reflect the initial and boundary conditions: T(x,0) = T∞ ∂ T −k x=0 = ∂ x ∂ T −k x=L = h  T ( L,t ) − T∞  − q′′o ∂ x uniform initial temperature insulated at x = surface energy balance at x = L PROBLEM 5.3 KNOWN: Microwave and radiant heating conditions for a slab of beef FIND: Sketch temperature distributions at specific times during heating and cooling SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in x, (2) Uniform internal heat generation for microwave, (3) Uniform surface heating for radiant oven, (4) Heat loss from surface of meat to surroundings is negligible during the heating process, (5) Symmetry about midplane ANALYSIS: COMMENTS: (1) With uniform generation and negligible surface heat loss, the temperature distribution remains nearly uniform during microwave heating During the subsequent surface cooling, the maximum temperature is at the midplane (2) The interior of the meat is heated by conduction from the hotter surfaces during radiant heating, and the lowest temperature is at the midplane The situation is reversed shortly after cooling begins, and the maximum temperature is at the midplane PROBLEM 5.4 KNOWN: Plate initially at a uniform temperature Ti is suddenly subjected to convection process (T∞,h) on both surfaces After elapsed time to, plate is insulated on both surfaces FIND: (a) Assuming Bi >> 1, sketch on T - x coordinates: initial and steady-state (t → ∞) temperature distributions, T(x,to) and distributions for two intermediate times to < t < ∞, (b) Sketch on T - t coordinates midplane and surface temperature histories, (c) Repeat parts (a) and (b) assuming Bi to, (5) T(0, t < to) < T∞ ANALYSIS: (a,b) With Bi >> 1, appreciable temperature gradients exist in the plate following exposure to the heating process On T-x coordinates: (1) initial, uniform temperature, (2) steady-state conditions when t → ∞, (3) distribution at to just before plate is covered with insulation, (4) gradients are always zero (symmetry), and (5) when t > to (dashed lines) gradients approach zero everywhere (c) If Bi