CBE 150A – Transport Spring Semester 2014 Non-Steady State Conduction

CBE 150A – Transport Spring Semester 2014 Goals : By the end of today’s lecture, you should be able to:  define the mechanisms for non-steady state conduction  determine the time required to transfer heat to and from:  flat plates  cylinders  spheres  describe the difference between constant surface combined convection and conduction. Non-steady State Conduction

CBE 150A – Transport Spring Semester 2014 Outline: I.Conduction for a constant boundary surface temperature Flat plate Cylinder Sphere II.Conduction for a rate based boundary temperature Flat plate Cylinder Sphere

CBE 150A – Transport Spring Semester 2014 Heat Flow TsTs TsTs dx Infinitely long solid slab (no end effects) (constant surface temperature) Heat Balance 2s

CBE 150A – Transport Spring Semester 2014 Where:  = thermal diffusivity = k/  c p Divide by  c p A dx dt to yield: Boundary Conditions:

CBE 150A – Transport Spring Semester 2014 Integrated Solution: Where:T s = constant average temperature of surface T a = initial temperature of slab T b = average temperature of the slab at time t F o = Fourier number =  t T /s 2  = thermal diffusivity = k/  c p t T time for heating s = one-half slab thickness a 1 = (  /2) 2

CBE 150A – Transport Spring Semester 2014 Neglect all but first term (for F o greater than 0.1) and get:

CBE 150A – Transport Spring Semester 2014 For infinitely long (no end effects) cylinder: Where: F o =  t T / r m 2

CBE 150A – Transport Spring Semester 2014 For a sphere: Where: F o =  t T / r m 2

CBE 150A – Transport Spring Semester 2014 Constant surface temperature plot Figure 10.5 Average temperatures during unsteady-state heating or cooling of a large slab, and infinitely long cylinder, or a sphere.

CBE 150A – Transport Spring Semester 2014 A sphere – heat transfer at boundary function of convective rate TsTs TfTf

CBE 150A – Transport Spring Semester 2014 Biot number ( Bi) = convection / conduction Flat plate Cylinder and sphere

CBE 150A – Transport Spring Semester 2014 For a sphere at low Biot number: Assuming an effective internal coefficient and an overall heat-transfer coefficient

CBE 150A – Transport Spring Semester 2014 Conductive / convective mechanism plot Figure 10.7 Change with time of the average temperature of a slab with external convective resistance.

CBE 150A – Transport Spring Semester 2014 Conductive / convective mechanism plot Figure 10.8 Change with time of the average temperature of a sphere with external convective resistance.

CBE 150A – Transport Spring Semester 2014 Solid Semi-infinite Solid TsTs T at time t and position x x Solid

CBE 150A – Transport Spring Semester 2014 Semi-infinite Solid TsTs T at time t x Solid Graphical solution to preceding equation

CBE 150A – Transport Spring Semester 2014 Problem Solution Matrix Problem Statement Steady State Calculate U,  T,Q,A Non-Steady State Constant Surface (T s ) or Convective Film (T f ) T Avg or T Position Resources Fig Fig Resources Fig TsTs TfTf Calculate U o or h o Resources Fig. (b-g ) T Avg Fig Fig Eqn T Position T Avg T Position T Avg or T Position Geometry (sphere, slab, cylinder)

CBE 150A – Transport Spring Semester 2014 Ten Minute Problem - The Thanksgiving Turducken I am cooking a 20 lb turducken (a turkey - stuffed with a duck - stuffed with a chicken – stuffed with stuffing – see photo below) for Thanksgiving dinner. How long will it take to cook ??? Initial temperature (T) of turducken on my kitchen counter = 70 F T oven = 350 F T of stuffing for a "done" turducken = 165 F External heat transfer coefficient for my Magic Chef natural circulating oven = 0.40 BTU / hr ft 2 F Assume the turducken is a fat thing that approximates a spherical geometry. Volume = 4/3  r 3 Surface area = 4  r 2 Effective density of turducken = 65 lb/ ft 3 Effective heat capacity of turducken = 0.83 BTU / lb F Thermal conductivity of turducken = 0.35 BTU / ft hr F

CBE 150A – Transport Spring Semester 2014