NonSteady State Conduction CBE 150 A Transport Spring

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

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

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

Divide by r cp A dx dt to yield: Where: a = thermal diffusivity = k/rcp Boundary Conditions: CBE 150 A – Transport Spring Semester 2014

Integrated Solution: Where: Ts = constant average temperature of surface Ta = initial temperature of slab Tb = average temperature of the slab at time t Fo = Fourier number = a t. T/s 2 a = thermal diffusivity = k/rcp t. T time for heating s = one-half slab thickness a 1 = (p/2)2 CBE 150 A – Transport Spring Semester 2014

Neglect all but first term (for Fo greater than 0. 1) and get: CBE 150 A – Transport Spring Semester 2014

For infinitely long (no end effects) cylinder: Where: Fo = a t. T / rm 2 CBE 150 A – Transport Spring Semester 2014

For a sphere: Where: Fo = a t. T / rm 2 CBE 150 A – 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 150 A – Transport Spring Semester 2014

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

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

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

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

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

Problem Solution Matrix Geometry Non-Steady State (sphere, slab, cylinder) Problem Statement Steady State Calculate U, DT, Q, A Constant Surface (Ts) or Convective Film (Tf) Ts TAvg or T Position Tf T Position Calculate TAvg Uo or ho TAvg or T Position Resources Fig. 11. 1. 2 TAvg T Position Resources Fig. (b-g) CBE 150 A – Transport Fig. 10. 7 Fig. 10. 8 Eqn. 10. 32 Fig. 10. 5 Resources Fig. 11. 1. 3 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 p r 3 Surface area = 4 p 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 150 A – Transport Spring Semester 2014

CBE 150 A – Transport Spring Semester 2014

CBE 150 A – Transport Spring Semester 2014