Unsteady-State
Heat Transfer
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
Simplified geometrical shapes
• Sphere
• Infinite cylinder
• Infinite slab
Example
• Food pasteurization
• Food sterilization
• Food refrigeration/chilling/cooling
Partial differential equation
• For a one-dimensional case,
temperature is a function of two
independent variables, time and
location :
Thermal diffusivity or 
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
Conductive resistance inside solid
Convective resistance in fluid layer
Internal resistance to heat transfer
External resistance to heat transfer
NBi
=
D/k = hD
1/h
k
D = characteristic dimension
= NBi (Biot number)
= NBi
Three cases for unsteady-state heat
transfer
• NBi < 0.1 : negligible internal resistance to heat
transfer
• 0.1 < NBi < 40 : finite internal and surface
resistance to heat transfer
• NBi > 40 : negligible surface resistance to heat
transfer
Negligible internal resistance to heat
transfer
NBi < 0.1
Example
Finite internal and surface resistance
to heat transfer
0.1 <NBi < 40
Solution
• Temperature-time charts
– Dimensionless number
K t
t
Fourier number = NFo =
= 2
2
Cp D
D
NBi = D/k
= hD
1/h
k
Ta–T
= temperature ratio
Ta–Ti
(Ta–T)
(Ta–Ti)
1/NBi
NFo = t / D2
(Ta–T)
(Ta–Ti)
1/NBi
NFo = t / D2
(Ta–T)
(Ta–Ti)
1/NBi
NFo = t / D2
• For calculation of temperature at any position
of the object, Gurney-Lurie Chart can be used.
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.
• As stated earlier the X term the time (t) and the thermal
diffusivity (α) 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
Negligible surface resistance to heat
transfer
NBi > 40
• Use temperature-time chart.
• The lines for k/hD or 1/NBi = 0 represent
negligible surface resistance to heat transfer
(Ta–T)
(Ta–Ti)
1/NBi
NFo = t / D2
Finite objects
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.
(Ta–T)
(Ta–Ti)
1/NBi
TR = 0.077
1/NBi = 0.237
NFo = t / D2
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.
Finite objects
(Ta–T)
(Ta–Ti)
1/NBi
NFO = 0.118
1/NBi = 0.004
NFo = t / D2
(Ta–T)
(Ta–Ti)
1/NBi
NFo = t / D2
NFO = 0.064
1/NBi = 0.03
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.