Unsteady-State Heat Transfer

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Unsteady-StateHeat Transfer
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Unsteady state or transient heat transfer

• Phase of heating and cooling process when the
  temperature is changing with time
• During this phase, temperature is a function of
  both location and time.
• Steady state temperature varies only with
  location

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Simplified geometrical shapes

• Sphere
• Infinite cylinder
• Infinite slab

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Example

• Food pasteurization
• Food sterilization
• Food refrigeration/chilling/cooling

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Partial differential equation

• For a one-dimensional case, temperature is a
  function of two independent variables, time and
  location

Thermal diffusivity or ??
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Important of external versus internal resistance
to heat transfer

• During unsteady state heating period, temperature
  in side a solid object (initially at a uniform
  temp.) will vary with location and time. Assuming
  the location of interest is at center of solid,
  heat transfer from to fluid to center will
  encounter two resistance

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Conductive resistance inside solid Convective
resistance in fluid layer
NBi (Biot number)
Internal resistance to heat transfer External
resistance to heat transfer
NBi
D characteristic dimension
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Three cases for unsteady-state heat transfer

• NBi lt 0.1 negligible internal resistance to
  heat transfer
• 0.1 lt NBi lt 40 finite internal and surface
  resistance to heat transfer
• NBi gt 40 negligible surface resistance to heat
  transfer

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Negligible internal resistance to heat transfer

• NBi lt 0.1

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Example
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Finite internal and surface resistance to heat
transfer

• 0.1 ltNBi lt 40

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Solution

• Temperature-time charts
• Dimensionless number

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1/NBi
NFo ?t / D2
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• For calculation of temperature at any position of
  the object, Gurney-Lurie Chart can be used.

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Gurney-Lurie Chart

• The chart shows how four different dimensionless
  groups depend on each other.
• For any given values of three of the groups the
  fourth can be read of the chart.

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• As stated earlier the X term the time (t) and the
  thermal diffusivity (a) is being divided by the
  radius (r) squared. In short n stands for the
  "depth", that is the length (x) divided by the
  total length (x0) or for cylinders and spheres
  the radius (r). The last term m is the
  relationship between the thermal

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Negligible surface resistance to heat transfer

• NBi gt 40

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• Use temperature-time chart.
• The lines for k/hD or 1/NBi 0 represent
  negligible surface resistance to heat transfer

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Finite objects
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Example 1

• Estimate the time when temperature at the
  geometric center of a 6 cm diameter apple held in
  2?C water stream reaches 3?C. The initial uniform
  temperature of the apple is 15?C. The convective
  heat transfer coefficient in water surrounding
  the apple is 50 W/m2?C. The properties of the
  apple are thermal conductivity k 0.355 W/m?C,
  specific heat Cp 3.6 kJ/kg?C, and density 820
  kg/m3.

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TR 0.077
1/NBi 0.237
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Example 2

• Estimate the temperature at the geometric center
  of a food product contained in a 303X406 can
  exposed to boiling water at 100?C for 30 min. The
  product is assumed to heat and cool by
  conduction. The initial uniform temperature of
  product is 35?C. The properties of the food are
  thermal conductivity k 0.34 W/m?C, specific
  heat Cp 3.5 kJ/kg?C, and density ? 900 kg/m3.
  The convective heat transfer coefficient for
  boiling water is estimated to be 2000 W/m2?C.

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Finite objects
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1/NBi 0.004
NFO 0.118
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1/NBi
NFo ?t / D2
1/NBi 0.03
NFO 0.064
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Example 3

• A rectangular slab of butter which is 46.2 mm
  thick at a temperature of 4.4?C in a cooler is
  removed and placed in an environment at 23.9?C.
  The sides and bottom of the butter container can
  be considered to be insulated by the container
  side walls. The flat top surface of the butter is
  exposed to the environment. The convective heat
  transfer coefficient is constant and is 8.52
  W/m2K. Calculate the temperature in the butter at
  the surface, at 25.4 mm below the surface, and at
  46.2 mm below the surface at the insulated bottom
  after 5 h of expose.
• The physical properties of butter are thermal
  conductivity k 0.197 W/m.K, specific heat Cp
  2.30 kJ/kg.K, and density ? 998 kg/m3.

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