18MEC202T- HEAT AND MASS TRANSFER
UNIT-1
General conduction equation,
boundary and initial conditions
Heat conduction
• The rate of heat conduction in a specified direction is proportional to the
temperature gradient, which is the change in temperature per unit
length in that direction.
• Heat conduction in a medium, in general, is three-dimensional and time
dependent. That is, T T(x, y, z, t) and the temperature in a medium
varies with position as well as time.
• Heat conduction in a medium is said to be steady when the temperature
does not vary with time, and unsteady or transient when it does.
• Heat conduction in a medium is said to be one-dimensional when
conduction is significant in one dimension only and negligible in the
other two dimensions, two-dimensional when conduction in the third
dimension is negligible, and three-dimensional when conduction in all
dimensions is significant.
Heat conduction equation:
• The governing differential equation for heat conduction in developed in
rectangular, cylindrical, and spherical coordinate systems.
Ref: Yunus A. Çengel
Rectangular Coordinates
Consider a small rectangular element of length x, width y,
and height z, as shown in Figure. Assume the density of
the body is and the specific heat is C.
An energy balance on this element during a small time
interval t can be expressed as
REF: Yunus A. Çengel
Eqn .(1)
The volume of the element is Velement= ΔxΔyΔz.
The change in the energy content of the element and the rate of
heat generation within the element can be expressed as
Eqn. (2)
Substituting eqn. (2) in (1) we get
Eqn. (3)
Dividing eqn (3) by ΔxΔyΔz gives
Ref: Yunus A. Çengel
Eqn. (4)
Noting that the heat transfer areas of the element for heat
conduction in the x, y, and z directions are Ax= Δy Δz, Ay= Δx
Δz, and Az= Δx Δy, respectively, and taking the limit as x, y, z
and t → 0 yields
Eqn. (5)
since, from the definition of the derivative and Fourier’s
law of heat conduction,
Eqn. (6)
Ref: Yunus A. Çengel
Equation 5 is the general heat conduction equation in
rectangular coordinates.
In the case of constant thermal conductivity, it reduces to
Eqn. (7)
where the property k/ ᵨC is again the thermal diffusivity of the
material.
Equation 7 is known as the Fourier-Biot equation, and it reduces
to these forms under specified conditions:
Eqn. (8)
Ref: Yunus A. Çengel
In Cylindrical coordinates,
Eqn. (9)
In Spherical coordinates,
Eqn. (10)
Ref: Yunus A. Çengel
BOUNDARY AND INITIAL CONDITIONS
• The differential equations do not incorporate any
information related to the conditions on the surfaces
such as the surface temperature or a specified heat
flux.
• The mathematical expressions of the thermal conditions
at the boundaries are called the boundary conditions.
• A condition, which is usually specified at time t= 0, is
called the initial condition, which is a mathematical
expression for the temperature distribution of the
medium initially.
Ref: Yunus A. Çengel
•In rectangular coordinates, the initial condition can
be specified in the general form as
Eqn. (11)
where the function f(x, y, z) represents the
temperature distribution throughout the medium
at time t = 0.
•When the medium is initially at a uniform
temperature of Ti, the initial condition of Eq. 11 can be
expressed as T(x, y, z, 0) = Ti.
• Under steady conditions, the heat conduction
equation does not involve any time derivatives, and
thus we do not need to specify an initial condition.
Ref: Yunus A. Çengel
Boundary conditions most commonly encountered in
practice are as follows:
• specified temperature
• specified heat flux
• Convection
• radiation boundary conditions.
Ref: Yunus A. Çengel
1. Specified Temperature Boundary Condition
For one-dimensional heat transfer through a plane wall of
thickness L, for example, the specified temperature
boundary conditions can be expressed as in the above
figure.
• where T1 and T2 are the specified temperatures at surfaces at x=0
and x= L, respectively. The specified temperatures can be
constant, which is the case for steady heat conduction, or may
vary with time.
Ref: Yunus A. Çengel
2. Specified Heat Flux Boundary Condition
• When there is sufficient information about energy
interactions at a surface, it may be possible to
determine the rate of heat transfer and thus the heat
flux q· (heat transfer rate per unit surface area,
W/m2) on that surface, and this information can be
used as one of the boundary conditions.
• The heat flux in the positive x-direction anywhere in the medium,
including the boundaries, can be expressed by Fourier’s law of heat
conduction as
• The sign of the specified heat flux is determined by inspection: positive if
the heat flux is in the positive direction of the coordinate axis, and negative
if it is in the opposite direction.
Ref: Yunus A. Çengel
Special Case: Insulated Boundary
• Heat transfer through a properly insulated
surface can be taken to be zero since adequate
insulation reduces heat transfer through a surface
to negligible levels.
• Therefore, a well-insulated surface can be
modeled as a surface with a specified heat flux of
zero.
• Then the boundary condition on a perfectly
insulated surface (at x= 0, for example) can be
expressed as
Ref: Yunus A. Çengel
Special Case: Thermal Symmetry
• The heat transfer problem in this plate will possess
thermal symmetry about the center plane at x= L/2.
• Also, the direction of heat flow at any point in the plate
will be toward the surface closer to the point, and there
will be no heat flow across the center plane.
• Therefore, the center plane can be viewed as an
insulated surface, and the thermal condition at this
plane of symmetry can be expressed as
Ref: Yunus A. Çengel
3. Convection Boundary Condition
The convection boundary condition is based
on a surface energy balance expressed as
For one-dimensional heat transfer in the x-
direction in a plate of thickness L,
the convection boundary conditions on both
surfaces can be expressed as
where h1 and h2 are the convection heat transfer coefficients and T1 and T2
are the temperatures of the surrounding mediums on the two sides of the
plate, Ref: Yunus A. Çengel
4. Radiation Boundary Condition
the radiation boundary condition on a surface
can be expressed as
For one-dimensional heat transfer in the x-direction in
a plate of thickness L, the radiation boundary
conditions on both surfaces can be expressed as
Ref: Yunus A. Çengel
Tutorial on boundary condition:
1. Consider a large plane wall of thickness L 0.2 m, thermal conductivity k =1.2
W/m · °C, and surface area A= 15 m2. The two sides of the wall are
maintained at constant temperatures of T 1 = 120°C and T2 = 50°C,
respectively, as shown in Figure. Determine (a) the variation of temperature
within the wall and the value of temperature at x = 0.1 m and (b) the rate of
heat conduction through the wall under steady conditions.
Solution:
Taking the direction normal to the surface of the wall to be
the x-direction, the differential equation for this problem
can be expressed as
with boundary conditions
Integrating the differential equation once with respect to x yields
Ref: Yunus A. Çengel
Integrating one more time, we obtain
Substituting the given information, the value of the temperature at x = 0.1 m
is determined to be
Ref: Yunus A. Çengel
(b) The rate of heat conduction anywhere in the wall is determined from
Fourier’s law to be
The numerical value of the rate of heat conduction through the wall is
determined by substituting the given values to be
Ref: Yunus A. Çengel
References
Yunus A. Çengel, Afshin J. Ghajar “Heat and Mass
Transfer”, Tata McGraw Hill Education, 2017.
