Problem 1.1
Heat is removed from the surface by convection. Therefore, Newton's law of cooling is
applicable.
Ambient temperature and heat transfer coefficient are uniform
Surface temperature varies along the rectangle.
Problem 1.2
Heat is removed from the surface by convection. Therefore, Newton's law of cooling
may be helpful.
Ambient temperature and surface temperature are uniform.
Surface area and heat transfer coefficient vary along the triangle.
Problem 1.3
Heat flux leaving the surface is specified (fixed).
Heat loss from the surface is by convection and radiation.
Convection is described by Newton's law of cooling.
Changing the heat transfer coefficient affects temperature distribution.
Surface temperature decreases as the heat transfer coefficient is increased.
Surface temperature gradient is described by Fourier’s law
Ambient temperature is constant.
Problem 1.4
Metabolic heat leaves body at the skin by convection and radiation.
Convection is described by Newton's law of cooling.
Fanning increases the heat transfer coefficient and affects temperature distribution,
including surface temperature.
Surface temperature decreases as the heat transfer coefficient is increased.
Surface temperature is described by Newton’s law of cooling.
) Ambient temperature is constant.
Problem 1.5
Melting rate of ice depends on the rate of heat added at the surface.
Heat is added to the ice from the water by convection.
Newton's law of cooling is applicable.
Stirring increases surface temperature gradient and the heat transfer coefficient. An
increase in gradient or h increases the rate of heat transfer.
Surface temperature remains constant equal to the melting temperature of ice.
Water temperature is constant.
Problem 1.6
This problem is described by cylindrical coordinates.
0.
For parallel streamlines v r v
Axial velocity is independent of axial and angular distance.
Problem 1.7
This problem is described by cylindrical coordinates.
Streamlines are concentric circles. Thus the velocity component in the radial direction
vanishes ( v r 0 ).
For one-dimensional flow there is no motion in the z-direction ( v z
The
0 ).
-velocity component, v , depends on distance r and time t.
Problem 1.8
This problem is described by Cartesian coordinates.
For parallel streamlines the y-velocity component v
0.
For one-dimensional flow there is no motion in the z-direction (w = 0).
The x-velocity component depends on distance y and time t.
Problem 1.9
This problem is described by Cartesian coordinates.
For parallel streamlines the y-velocity component v
0.
For one-dimensional flow there is no motion in the z-direction (w = 0).
The x-velocity component depends on distance y only.
Problem 1.10
Heat flux leaving the surface is specified (fixed).
Heat loss from the surface is by convection and radiation
Convection is described by Newton's law of cooling.
Changing the heat transfer coefficient affects temperature distribution.
Surface temperature decreases as the heat transfer coefficient is increased.
Surface temperature gradient is described by Fourier’s law
Ambient temperature is constant.
Problem 1.11
Heat is removed from the surface by convection. Therefore, Newton's law of cooling is
applicable.
Ambient temperature and heat transfer coefficient are uniform.
Surface temperature varies along the area.
The area varies with distance x.
Problem 2.2
The fluid is incompressible.
Radial and tangential velocity components are zero.
Streamlines are parallel.
Cylindrical geometry.
Problem 2.3
The fluid is incompressible.
axial velocity is invariant with axial distance.
Plates are parallel.
Cartesian geometry.
Problem 2.4
The fluid is incompressible.
Radial and tangential velocity components are zero.
Streamlines are parallel.
Cylindrical geometry.
Problem 2.5
Shearing stresses are tangential surface forces.
xy
and
yx
are shearing stresses in a Cartesian coordinate system.
Tangential forces on an element result in angular rotation of the element.
If the net external torque on an element is zero its angular acceleration will vanish.
Problem 2.6
Properties are constant.
Cartesian coordinates.
Parallel streamlines: no velocity component in the y-direction.
Axial flow: no velocity component in the z-direction.
The Navier-Stokes equations give the three momentum equations.
Problem 2.7
Properties are constant.
Cylindrical coordinates.
Parallel streamlines: no velocity component in the r-direction.
Axial flow: no velocity component in the
-direction.
No variation in the -direction. The Navier-Stokes equations give the three momentum
equations.
Problem 2.8
Properties are constant.
Cartesian coordinates.
Two dimensional flow (no velocity component in the z-direction
The Navier-Stokes equations give two momentum equations.
Problem 2.9
Properties are constant.
Cylindrical coordinates.
Two dimensional flow (no velocity component in the
-direction.
The Navier-Stokes equations give two momentum equations.
Problem 2.10
Motion in energy consideration is represented by velocity components.
Fluid nature is represented by fluid properties.
Problem 2.11
Properties are constant.
Cartesian coordinates.
Parallel streamlines: no velocity component in the y-direction.
Axial flow: no velocity component in the z-direction.
Problem 2.12
Properties are constant.
Cartesian coordinates.
Parallel streamlines: no velocity component in the y-direction.
Axial flow: no velocity component in the z-direction.
The fluid is an ideal gas.
Problem 2.13
This is a two-dimensional free convection problem.
The flow is due to gravity.
The flow is governed by the momentum and energy equations. Thus the governing
equations are the Navier-Stokes equations of motion and the energy equation.
The geometry is Cartesian.
Problem 2.15
The flow is due to gravity.
For parallel streamlines the velocity component v = 0 in the y-direction.
Pressure at the free surface is uniform (atmospheric).
Properties are constant.
The geometry is Cartesian.
Problem 2.16
This is a forced convection problem.
Flow properties (density and viscosity) are constant.
Upstream conditions are uniform (symmetrical)
The velocity vanishes at both wedge surfaces (symmetrical).
Surface temperature is asymmetric.
Flow field for constant property fluids is governed by the Navier-Stokes and continuity
equations.
If the governing equations are independent of temperature, the velocity distribution over
the wedge should be symmetrical with respect to x.
The geometry is Cartesian.
Problem 2.18
The geometry is Cartesian.
Properties are constant.
Axial flow (no motion in the z-direction).
Parallel streamlines means that the normal velocity component is zero.
Specified flux at the lower plate and specified temperature at the upper plate.
Problem 2.19
The geometry is cylindrical.
No variation in the axial and angular directions.
Properties are constant.
Problem 2.20
The geometry is cylindrical.
No variation in the angular direction.
Properties are constant.
Parallel streamlines means that the radial velocity component is zero.
Problem 2.21
The geometry is cylindrical. (ii)
No variation in the axial and angular directions.
Properties are constant.
Problem 2.22
This is a forced convection problem.
The same fluid flows over both spheres.
Sphere diameter and free stream velocity affect the Reynolds number which in turn affect
the heat transfer coefficient.
Problem 2.23
This is a free convection problem.
The average heat transfer coefficient h depends on the vertical length L of the plate.
L appears in the Nusselt number as well as the Grashof number.
Problem 2.24
This is a forced convection problem.
The same fluid flows over both spheres.
Sphere diameter and free stream velocity affect the Reynolds number which in turn affect
the heat transfer coefficient. (iv) Newton’s law of cooling gives the heat transfer
Problem 2.25
Dissipation is important when the Eckert number is high compared to unity.
If the ratio of dissipation to conduction is small compared to unity, it can be neglected.
Problem 2.26
The plate is infinite.
No changes take place in the axial direction (infinite plate).
This is a transient problem.
Constant properties.
Cartesian coordinates.
Problem 2.27
The plate is infinite.
No changes take place in the axial direction (infinite plate).
This is a transient problem.
Constant properties.
Cartesian coordinates.
Gravity is neglected. Thus there is no free convection.
The fluid is stationary.
Problem 3.1
Moving plate sets fluid in motion in the x-direction.
Since plates are infinite the flow field does not vary in the axial direction x.
The effect of pressure gradient is negligible.
The fluid is incompressible (constant density).
Use Cartesian coordinates.
Problem 3.2
Moving plate sets fluid in motion in the x-direction.
Since plates are infinite the flow field does not vary in the axial direction x.
The effect of pressure gradient must be included.
The fluid is incompressible.
Using Fourier’s law, Temperature distribution gives surface heat flux of the moving
plate.
Use Cartesian coordinates.
Problem 3.3
Moving plate sets fluid in motion in the x-direction.
Since plates are infinite the flow field does not vary in the axial direction x.
The fluid is incompressible (constant density).
Use Cartesian coordinates.
Problem 3.4
Moving plates set fluid in motion in the positive and negative x-direction.
Since plates are infinite the flow field does not vary in the axial direction x.
The fluid is incompressible (constant density).
The fluid is stationary at the center plane y = 0.
Symmetry dictates that no heat is conducted through the center plane.
Use Cartesian coordinates.
Problem 3.5
Fluid motion is driven by axial pressure drop.
For a very long tube the flow field does not vary in the axial direction z.
The fluid is incompressible (constant density).
Heat is generated due to viscous dissipation. It is removed from the fluid by convection at
the surface.
The Nusselt number is a dimensionless heat transfer coefficient.
To determine surface heat flux and heat transfer coefficient requires the determination of
temperature distribution.
Temperature distribution depends on the velocity distribution.
Use cylindrical coordinates.
Problem 3.6
Fluid motion is driven by axial pressure drop.
For a very long tube the flow field does not vary in the axial direction z.
The fluid is incompressible (constant density).
Use cylindrical coordinates.
Problem 3.7
Fluid motion is driven by axial motion of the rod. Thus motion is not due to pressure
gradient.
For a very long tube the flow field does not vary in the axial direction z.
The fluid is incompressible (constant density).
Heat is generated due to viscous dissipation. It is removed from the fluid by conduction at
the surface.
The Nusselt number is a dimensionless heat transfer coefficient.
To determine the heat transfer coefficient require the determination of temperature
distribution.
Temperature distribution depends on the velocity distribution.
Use cylindrical coordinates.
Problem 3.8
Fluid motion is driven by gravity.
No velocity and temperature variation in the axial direction.
The fluid is incompressible (constant density).
Heat is generated due to viscous dissipation.
Temperature distribution depends on the velocity distribution.
Use Cartesian coordinates.
Problem 3.9
Fluid motion is driven by gravity.
No velocity and temperature variation in the axial direction.
The fluid is incompressible (constant density).
Heat is generated due to viscous dissipation.
Temperature distribution depends on the velocity distribution.
the inclined surface is at specified temperature and the free surface exchanges heat by
convection with the ambient.
Use Cartesian coordinates.
Problem 6.10
This is an internal forced convection problem.
The channel has a rectangular cross section.
Surface temperature is uniform.
The Reynolds and Peclet numbers should be checked to establish if the flow is laminar or
turbulent and if entrance effects can be neglected.
Channel length is unknown.
The fluid is air.
Problem 3.11
Fluid motion is driven by shaft rotation
The housing is stationary.
Axial variation in velocity and temperature are negligible for a very long shaft.
Velocity and temperature do not vary with angular position.
The fluid is incompressible (constant density).
Heat generated by viscous dissipation is removed from the oil at the housing.
No heat is conducted through the shaft.
The maximum temperature occurs at the shaft.
Heat flux at the housing is determined from temperature distribution and Fourier’s law of
conduction.
Use cylindrical coordinates.
Problem 3.12
Fluid motion is driven by sleeve rotation
The shaft is stationary.
Axial variation in velocity and temperature are negligible for a very long shaft.
Velocity and temperature do not vary with angular position.
The fluid is incompressible (constant density).
Heat generated by viscous dissipation is removed from the oil at the housing.
No heat is conducted through the shaft.
The maximum temperature occurs at the shaft. (ix) Use cylindrical coordinates.
Problem 3.13
Fluid motion is driven by shaft rotation
Axial variation in velocity and temperature are negligible for a very long shaft.
Velocity, pressure and temperature do not vary with angular position.
The fluid is incompressible (constant density).
Heat generated by viscous dissipation is conducted radially.
The determination of surface temperature and heat flux requires the determination of
temperature distribution in the rotating fluid.
Use cylindrical coordinates.
Problem 3.14
Axial pressure gradient sets fluid in motion.
The fluid is incompressible.
The flow field is determined by solving the continuity and Navier-Stokes equations.
Energy equation gives the temperature distribution.
Fourier’s law and temperature distribution give surface heat flux.
Axial variation of temperature is neglected.
Problem 4.3
This is forced convection flow over a streamlined body.
Viscous (velocity) boundary layer approximations can be made if the Reynolds number
Rex > 100.
Thermal (temperature) boundary layer approximations can be made if the Peclet number
Pex = Rex Pr > 100.
The Reynolds number decreases as the distance along the plate is decreased.
Problem 4.4
The surface is streamlined.
The fluid is water.
Inertia and viscous effects can be estimated using scaling.
If a viscous term is small compared to inertia, it can be neglected.
Properties should be evaluated at the film temperature T f
(Ts
T ) / 2.
Problem 4.5
The surface is streamlined.
The fluid is water.
Convection and conduction effects can be estimated using scaling.
If a conduction term is small compared to convection, it can be neglected.
The scale for
t
/ L depends on whether
t
or
t
.
Properties should be evaluated at the film temperature T f
(Ts
T ) / 2.
Problem 4.6
The fluid is air.
Dissipation and conduction can be estimated using scaling.
Dissipation is negligible if the Eckert number is small compared to unity.
Problem 4.7
The surface is streamlined.
The fluid is air.
Problem 4.9
This is a forced convection problem over a flat plate.
At the edge of the thermal boundary layer, the axial velocity is u V .
Blasius solution gives the distribution of the velocity components u(x,y) and v(x,y).
Scaling gives an estimate of v(x,y).
Problem 4.11
This is a laminar boundary layer flow problem.
Blasius solution gives the velocity distribution for the flow over a semi-infinite flat plate.
(iii) A solution for the boundary layer thickness depends on how the thickness is defined.
Problem 4.12
Since the flow within the boundary layer is two-dimensional the vertical velocity
component does not vanish. Thus stream lines are not parallel.
Blasius solution is valid for laminar boundary layer flow over a semi-infinite plate.
The transition Reynolds number from laminar to turbulent flow is 5 105 .
Boundary layer approximations are valid if the Reynolds number is greater than 100.
Problem 4.13
This is an external flow problem over a flat plate.
Blasius’s solution for the velocity distribution and wall shearing stress is assumed to be
applicable.
Of interest is the value of the local stress at the leading edge of the plate.
Problem 4.14
This is an external flow problem over a flat plate.
The force needed to hold the plate in place is equal to the total shearing force by the fluid
on the plate.
Integration of wall shear over the surface gives the total shearing force.
Blasius’s solution for the velocity distribution and wall shearing stress is assumed to be
applicable.
Problem 4.16
This is an external forced convection problem for flow over a flat plate.
Of interest is the region where the upstream fluid reaches the leading edge of the plate.
The fluid is heated by the plate.
Heat from the plate is conducted through the fluid in all directions.
Pohlhausen’s solution assumes that heat is not conducted upstream from the plate and
therefore fluid temperature at the leading edge is the same as upstream temperature.
Problem 4.18
This is a forced convection problem over a flat plate.
At the edge of the thermal boundary layer, fluid temperature is T
T .
Pohlhausen’s solution gives the temperature distribution in the boundary layer.
The thermal boundary layer thickness
t
t
increases with distance from the leading edge.
depends on the Prandtl number.
Problem 4.19
This is an external forced convection problem for flow over a flat plate.
Pohlhausen’s solution for the temperature distribution and heat transfer coefficient is
assumed to be applicable.
Of interest is the value of the local heat flux at the leading edge of the plate.
Knowing the local transfer coefficient and using Newton’s law, gives the heat flux
Problem 4.20
This is an external forced convection problem for flow over a flat plate.
Pohlhausen’s solution for the temperature distribution is assumed to be applicable.
Of interest is the value of the normal temperature gradient at the surface.
Problem 4.22`
This is an external forced convection problem over two flat plates.
Both plates have the same surface area.
For flow over a flat plate, the heat transfer coefficient h decreases with distance from the
leading edge.
Since the length in the flow direction is not the same for the two plates, the average heat
transfer coefficient is not the same. It follows that the total heat transfer rate is not the
same.
The flow over a flat plate is laminar if the Reynolds number is less than 5 105.
Problem 4.23
This is an external forced convection problem for flow over a flat plate.
Pohlhausen’s solution for the temperature distribution and heat transfer coefficient is
assumed to be applicable.
Of interest is the value of the heat transfer rate from a section of the plate at a specified
location and of a given width.
Newton’s law of cooling gives the heat transfer rate.
Problem 4.24
This is an external forced convection problem for flow over a flat plate.
Of interest is the variation of the local heat transfer coefficient with free stream velocity
and distance from the leading edge.
Pohlhausen's solution applies to this problem.
Problem 4.25
This is an external flow problem.
At the edge of the thermal boundary layer, y
stream temperature. That is, T
T and T *
(T
t
, fluid temperature approaches free
Ts ) /(T
Ts ) 1 .
According to Pohlhausen's solution, Fig. 4.6, the thermal boundary layer thickness
depends on the Prandtl number, free stream velocity V , kinematic viscosity
and
location x.
Problem 4.26
This is an external forced convection problem for flow over a flat plate.
The Reynolds number and Peclet number should be checked to determine if the flow is
laminar and if boundary layer approximations are valid.
Pohlhausen's solution is applicable if 100 < Rex < 100 105 and Pex = Rex Pr > 100.
Thermal boundary layer thickness and heat transfer coefficient vary along the plate.
Newton’s law of cooling gives local heat flux. (vi) The fluid is water.
Problem 4.27
This is an external forced convection problem over a flat plate.
Increasing the free stream velocity, increases the average heat transfer coefficient. This in
turn causes surface temperature to drop.
Based on this observation, it is possible that the proposed plan will meet design
specification.
Since the Reynolds number at the downstream end of the package is less than 500,000, it
follows that the flow is laminar throughout.
Increasing the free stream velocity by a factor of 3, increases the Reynolds number by a
factor of 3 to 330,000. At this Reynolds number the flow is still laminar.
The power supplied to the package is dissipated into heat and transferred to the
surroundings from the surface.
Pohlhausen's solution can be applied to this problem.
The ambient fluid is unknown.
Problem 4.28
This is an external forced convection problem of flow over a flat plate.
Convection heat transfer from a surface can be determined using Newton’s law of
cooling.
The local heat transfer coefficient changes along the plate. The total heat transfer rate
can be determined using the average heat transfer coefficient.
For laminar flow, Pohlhausen's solution gives the heat transfer coefficient.
For two in-line fins heat transfer from the down stream fin is influenced by the upstream
fin. The further the two fins are apart the less the interference will be.
Problem 4.29
This is an external forced convection problem for flow over a flat plate.
Pohlhausen’s solution for the temperature distribution and heat transfer coefficient is
assumed to be applicable.
Knowing the heat transfer coefficient, the local Nusselt number can be determined.
the Newton’s law of cooling gives the heat transfer rate.
Pohlhausen’s solution gives the thermal boundary layer thickness.
Problem 4.30
This is an external forced convection problem of flow over a flat plate.
Convection heat transfer from a surface can be determined using Newton’s law of
cooling.
The local heat transfer coefficient changes along the plate.
For each triangle the area changes with distance along the plate.
The total heat transfer rate can be determined by integration along the length of each
triangle.
Pohlhausen's solution may be applicable to this problem.
Problem 4.31
This is an external forced convection problem of flow over a flat plate.
Heat transfer rate can be determined using Newton’s law of cooling.
The local heat transfer coefficient changes along the plate.
The area changes with distance along the plate.
The total heat transfer rate can be determined by integration along the length of the
triangle.
Pohlhausen's solution may be applicable to this problem.
Problem 4.32
This is an external forced convection problem of flow over a flat plate.
Heat transfer rate can be determined using Newton’s law of cooling.
The local heat transfer coefficient changes along the plate.
The area changes with distance along the plate.
The total heat transfer rate can be determined by integration along over the area of the
semi-circle.
Pohlhausen's solution gives the heat transfer coefficient.
Problem 4.33
This is an external forced convection problem of flow over a flat plate.
Heat transfer rate can be determined using Newton’s law of cooling.
The local heat transfer coefficient changes along the plate.
The area changes with distance along the plate.
The total heat transfer rate can be determined by integration along the length of the
triangle.
Pohlhausen's solution may be applicable to this problem.
Problem 4.34
This is an external forced convection problem of flow over a flat plate.
This problem involves determining the heat transfer rate from a circle tangent to the
leading edge of a plate
Heat transfer rate can be determined using Newton’s law of cooling.
The local heat transfer coefficient changes along the plate.
The area changes with distance along the plate.
The total heat transfer rate can be determined by integration along the length of the
triangle.
Pohlhausen's solution may be applicable to this problem.
Problem 4.36
The flow field for this boundary layer problem is simplified by assuming that the axial
velocity is uniform throughout the thermal boundary layer.
Since velocity distribution affects temperature distribution, the solution for the local
Nusselt number can be expected to differ from Pohlhausen’s solution.
The Nusselt number depends on the temperature gradient at the surface.
Problem 4.37
The flow field for this boundary layer problem is simplified by assuming that the axial
velocity varies linearly in the y-direction.
Since velocity distribution affects temperature distribution, the solution for the local
Nusselt number can be expected to differ from Pohlhausen’s solution.
The Nusselt number depends on the temperature gradient at the surface.
Problem 4.38
The flow and temperature fields for this boundary layer problem are simplified by
assuming that the axial velocity and temperature do not vary in the x-direction.
The heat transfer coefficient depends on the temperature gradient at the surface.
Temperature distribution depends on the flow field.
The effect of wall suction must be taken into consideration.
Problem 4.39
This is a forced convection flow over a plate with variable surface temperature.
The local heat flux is determined by Newton’s law of cooling.
The local heat transfer coefficient and surface temperature vary with distance along the
plate. The variation of surface temperature and heat transfer coefficient must be such that
Newton’s law gives uniform heat flux.
The local heat transfer coefficient is obtained from the local Nusselt number.
Problem 4.40
This is a forced convection flow over a plate with variable surface temperature.
The Reynolds number should be computed to determine if the flow is laminar or
turbulent.
The local heat transfer coefficient and surface temperature vary with distance along the
plate.
The local heat transfer coefficient is obtained from the solution to the local Nusselt
number.
The determination of the Nusselt number requires determining the temperature gradient
at the surface.
Problem 4.41
This is a forced convection flow over a plate with variable surface temperature.
The Reynolds number should be computed to determine if the flow is laminar or
turbulent.
Newton’s law of cooling gives the heat transfer rate from the plate.
The local heat transfer coefficient and surface temperature vary with distance along the
plate. Thus determining the total heat transfer rate requires integration of Newton’s law
along the plate.
The local heat transfer coefficient is obtained from the local Nusselt number.
Problem 4.42
This is an external forced convection problem of flow over a flat plate
Convection heat transfer from a surface can be determined using Newton’s law of
cooling.
The local heat transfer coefficient and surface temperature vary along the plate.
For each triangle the area varies with distance along the plate.
The total heat transfer rate can be determined by integration along the length of each
triangle.
Problem 4.43
This is a forced convection boundary layer flow over a wedge.
Wedge surface is maintained at uniform temperature.
The flow is laminar.
The fluid is air.
Similarity solution for the local Nusselt number is presented in Section 4.4.3.
The Nusselt number depends on the Reynolds number and the dimensionless temperature
gradient at the surface d (0) / d . (vii) Surface temperature gradient depends on wedge
angle.
Problem 4.44
This is a forced convection boundary layer flow over a wedge.
Wedge surface is maintained at uniform temperature.
The flow is laminar.
The average Nusselt number depends on the average heat transfer coefficient..
Similarity solution for the local heat transfer coefficient is presented in Section 4.4.3.
Problem 4.45
Newton’s law of cooling gives the heat transfer rate from a surface.
Total heat transfer from a surface depends on the average heat transfer coefficient h .
Both flat plate and wedge are maintained at uniform surface temperature.
Pohlhausen’s solution gives h for a flat plate.
Similarity solution for the local heat transfer coefficient for a wedge is presented in
Section 4.4.3.
Problem 4.46
The flow field for this boundary layer problem is simplified by assuming that axial
velocity within the thermal boundary layer is the same as that of the external flow.
Since velocity distribution affects temperature distribution, the solution for the local
Nusselt number differs from the exact case of Section 4.4.3.
The local Nusselt number depends the local heat transfer coefficient which depends on
the temperature gradient at the surface.
Problem 4.47
The flow field for this boundary layer problem is simplified by assuming that axial
velocity within the thermal boundary layer varies linearly with the normal distance.
Since velocity distribution affects temperature distribution, the solution for the local
Nusselt number differs from the exact case of Section 4.4.3.
The local Nusselt number depends on the local heat transfer coefficient which depends on
the temperature gradient at the surface.
Problem 5.1
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution
1 is assumed to be uniform, u
Fluid velocity for Pr
significant simplification.
V . This represents a
Problem 5.2
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
Fluid velocity for Pr
1 is assumed to be linear, u
V (y / ) .
Problem 5.3
The velocity is assumed to be uniform, u V , throughout the thermal boundary layer.
A leading section of length x o is unheated.
at x
xo , surface heat flux is uniform.
The determination of the Nusselt number requires the determination of the temperature
distribution.
Surface temperature is unknown.
The maximum surface temperature for a uniformly heated plate occurs at the trailing end.
Problem 5.4
The velocity distribution is known.
Surface temperature is uniform.
The determination of the Nusselt number requires the determination of the temperature
distribution.
Newton’s law of cooling gives the heat transfer rate. This requires knowing the local heat
transfer coefficient.
Problem 5.5
The velocity distribution is known
Total heat transfer is equal to heat flux times surface area.
Heat flux is given. However, the distance x = L at which
t
H / 2 is unknown.
Problem 5.6
The determination of the Nusselt number requires the determination of the velocity and
temperature distributions.
Velocity is assumed uniform.
Surface temperature is variable.
Newton’s law of cooling gives surface heat flux. This requires knowing the local heat
transfer coefficient.
Problem 5.7
The determination of the Nusselt number requires the determination of the velocity and
temperature distributions.
Velocity is assumed linear.
Surface temperature is variable.
Newton’s law of cooling gives surface heat flux. This requires knowing the local heat
transfer coefficient.
Problem 5.8
The determination of the Nusselt number requires the determination of the velocity and
temperature distributions.
Velocity is assumed uniform.
Surface temperature is variable.
Newton’s law of cooling gives surface heat flux. This requires knowing the local heat
transfer coefficient.
Problem 5.9
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
Surface heat flux is variable. It decreases with distance x.
Surface temperature is unknown.
Newton’s law of cooling gives surface temperature. This requires knowing the local heat
1.
transfer coefficient. (v) t /
Problem 5.10
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
1 is assumed to be uniform, u
Fluid velocity for Pr
significant simplification.
V . This represents a
Surface heat flux is variable. It increases with distance x.
Surface temperature is unknown. Since flux increases with x and heat transfer coefficient
decreases with x, surface temperature is expected to increase with x. Thus maximum
surface temperature is at the trailing end x = L.
Newton’s law of cooling gives surface temperature. This requires knowing the local heat
transfer coefficient.
Problem 5.11
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
1 is assumed to be uniform, u
Fluid velocity for Pr
significant simplification.
V . This represents a
Surface heat flux is variable. It increases with distance x.
Surface temperature is unknown. Since flux increases with x and heat transfer coefficient
decreases with x, surface temperature is expected to increase with x. Thus maximum
surface temperature is at the trailing end x = L.
Newton’s law of cooling gives surface temperature. This requires knowing the local heat
transfer coefficient.
Problem 5.12
This problem is described by cylindrical coordinates.
Velocity variation with y is negligible.
Conservation of mass requires that radial velocity decrease with radial distance r.
Surface temperature is uniform.
Problem 5.13
This problem is described by cylindrical coordinates.
Velocity variation with y is negligible.
Conservation of mass requires that radial velocity decrease with radial distance r.
Surface heat flux is uniform
Surface temperature is unknown.
Problem 5.14
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
1 is assumed to be uniform, u
Fluid velocity for Pr
significant simplification.
V . This represents a
The plate is porous.
Fluid is injected through the plate with uniform velocity.
The plate is maintained at uniform surface temperature.
A leading section of the plate is insulated.
Problem 5.15
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
1 is assumed to be uniform, u
Fluid velocity for Pr
significant simplification.
V . This represents a
The plate is porous.
Fluid is injected through the plate with uniform velocity.
The plate is heated with uniform surface flux
Surface temperature is unknown, (vii) A leading section of the plate is insulated.
Problem 5.16
There are two thermal boundary layers in this problem.
The upper and lower plates have different boundary conditions. Thus, temperature
distribution is not symmetrical.
The lower plate is at uniform temperature while heat is removed at uniform flux along the
upper plate.
Fluid velocity is assumed uniform throughout the channel.
Problem 6.1
This is an internal forced convection problem.
Scaling gives estimates of L h and Lt .
Exact solutions for L h and Lt are available for laminar flow through channels.
Exact solutions for Lt depend on channel geometry and surface boundary conditions.
Problem 6.2
This is an internal forced convection problem.
Scaling gives estimates of L h and
Exact solutions for L h and Lt are available for laminar flow through channels.
Exact solutions for Lt depend on channel geometry and surface boundary conditions.
Problem 6.3
This is an internal force convection problem.
The channel is a long tube.
The surface is maintained at a uniform temperature.
Since the tube section is far away from the entrance, the velocity and temperature can be
assumed fully developed.
Tube diameter, mean velocity and inlet, outlet and surface temperatures are known. The
length is unknown.
The fluid is air.
Problem 6.4
This is an internal force convection in a tube.
The surface is heated at uniform flux.
Surface temperature increases along the tube and is unknown.
The flow is assumed laminar and fully developed.
The heat transfer coefficient for fully developed flow through channels is constant.
According to Newton’s law of cooling, surface temperature is related to mean fluid
temperature, surface heat flux and heat transfer coefficient.
Problem 6.5
This is an internal force convection in a tube.
The surface is heated at uniform flux.
Surface temperature increases along the tube and is unknown.
The flow is assumed laminar and fully developed.
The heat transfer coefficient for fully developed flow through channels is constant.
According to Newton’s law of cooling, surface temperature is related to mean fluid
temperature, surface heat flux and heat transfer coefficient.
Problem 6.6
This is an internal forced convection problem in a tube.
The surface is heated at uniform flux.
Surface temperature changes along the tube and is unknown.
The Reynolds number should be checked to determine if the flow is laminar or turbulent.
If hydrodynamic and thermal entrance lengths are small compared to tube length, the
flow can be assumed fully developed throughout.
For fully developed flow, the heat transfer coefficient is uniform.
The length of the tube is unknown.
The fluid is water.
Problem 6.7
This is an internal force convection problem.
The channel is a tube.
The surface is maintained at a uniform temperature.
Entrance effect is important in this problem.
The average Nusselt number for a tube of length L depends on the average heat transfer
coefficient over the length.
Problem 6.8
This is an internal forced convection problem.
The fluid is heated at uniform wall flux.
Surface temperature changes with distance along the channel. It reaches a maximum
value at the outlet.
The Reynolds and Peclet numbers should be checked to establish if the flow is laminar or
turbulent and if this is an entrance or fully developed problem.
The channel has a square cross-section.
Application of Newton’s law of cooling at the outlet relates outlet temperature to surface
temperature, surface flux and heat transfer coefficient.
Application of conservation of energy gives a relationship between heat added, inlet
temperature, outlet temperature, specific heat and mass flow rate.
Problem 6.9
This is an internal forced convection problem in tubes.
The flow is laminar and fully developed.
The surface is maintained at uniform temperature.
All conditions are identical for two experiments except the flow rate through one is half
that of the other.
The total heat transfer rate depends on the outlet temperature.
Problem 6.10
This is an internal forced convection problem.
The channel has a rectangular cross section.
Surface temperature is uniform.
The Reynolds and Peclet numbers should be checked to establish if the flow is laminar or
turbulent and if entrance effects can be neglected.
Channel length is unknown.
The fluid is air.
Problem 6.11
This is an internal force convection problem.
The channel is a rectangular duct.
The surface is maintained at a uniform temperature.
The velocity and temperature are fully developed.
The Reynolds number should be checked to determine if the flow is laminar or turbulent.
Duct size, mean velocity and inlet, outlet and surface temperatures are known. The
length is unknown. (vii) Duct length depends on the heat transfer coefficient.
The fluid is water.
Problem 6.12
This is an internal forced convection problem in a channel.
The surface is heated at uniform flux.
Surface temperature changes along the channel. It reaches a maximum value at the outlet.
The Reynolds number should be checked to determine if the flow is laminar or turbulent.
Velocity and temperature profiles become fully developed far away from the inlet.
The heat transfer coefficient is uniform for fully developed flow.
The channel has a square cross section.
tube length is unknown. (ix) The fluid is air.
Problem 6.13
This is an internal forced convection problem in tubes.
The flow is laminar and fully developed.
The surface is maintained at uniform temperature.
All conditions are identical for two tubes except the diameter of one is twice that of the
other.
The total heat transfer in each tube depends on the outlet temperature.
Problem 6.14
This is an internal forced convection problem.
Equation (6.3) gives scaling estimate of the thermal entrance length.
Equation (6.20b) gives scaling estimate of the local Nusselt number.
The Graetz problem deals with laminar flow in the entrance of a tube at uniform surface
temperature.
Graetz solutions gives the thermal entrance length (distance to reach fully developed
temperature) and local Nusselt number.
Problem 6.15
This is an internal forced convection problem.
Equation (6.20b) gives scaling estimate of the local Nusselt number.
The Graetz problem deals with laminar flow in the entrance of a tube at uniform surface
temperature.
Problem 6.16
This is an internal forced convection problem in a tube.
The velocity is fully developed.
The temperature is developing.
Surface is maintained at uniform temperature.
The Reynolds number should be computed to establish if flow is laminar or turbulent.
Tube length is unknown.
The determination of tube length requires determining the heat transfer coefficient.
Problem 6.17
This is an internal forced convection problem in a tube.
The velocity is fully developed.
The temperature is developing.
Surface is maintained at uniform temperature.
The Reynolds number should be computed to establish if flow is laminar or turbulent.
Outlet mean temperature is unknown.
The determination of outlet temperature
coefficient.
requires determining the heat transfer
Since outlet temperature is unknown, air properties can not be determined. Thus a trial
and error procedure is needed to solve the problem.
Problem 6.18
This is an internal forced convection problem in a tube.
The velocity is fully developed and the temperature is developing.
The surface is heated with uniform flux.
The Reynolds number should be computed to establish if the flow is laminar or turbulent.
Compute thermal entrance length to determine if it can be neglected.
Surface temperature varies with distance from entrance. It is maximum at the outlet. Thus
surface temperature at the outlet is known.
Analysis of uniformly heated channels gives a relationship between local surface
temperature, heat flux and heat transfer coefficient.
The local heat transfer coefficient varies with distance form the inlet.
Knowing surface heat flux, the required power can be determined.
Newton’s law of cooling applied at the outlet gives outlet temperature.
Problem 6.19
This is an internal forced convection problem in a rectangular channel.
The velocity is fully developed and the temperature is developing.
The surface is maintained at uniform temperature.
The Reynolds number should be computed to establish if the flow is laminar or turbulent.
Compute entrance lengths to determine if they can be neglected
Surface flux varies with distance from entrance. It is minimum at outlet.
Newton’s law gives surface flux in terms of the local heat transfer coefficient h(x) and
the local mean temperature Tm (x) .
The local and average heat transfer coefficient decrease with distance form the inlet.
The local mean temperature depends on the local average heat transfer coefficient
h (x). (x) Surface temperature is unknown.
Problem 7.2
This is an external free convection problem over a vertical plate.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
The solution for laminar flow is given in Section 7.4
For laminar flow, Fig.7.2 gives the viscous boundary layer thickness
the thermal boundary layer thickness t .
and Fig. 7.3 gives
Newton’s law of cooling gives the heat transfer rate.
Equation (7.23) gives the average heat transfer coefficient h . (vii) The fluid is water.
Problem 7.3
This is an external free convection problem for flow over a vertical plate.
Laminar flow solution for temperature distribution for a plate at uniform surface
temperature is given in Fig. 7.3 .
The dimensionless temperature gradient at the surface is given in Table 7.1.
The solution depends on the Prandtl number.
Problem 7.4
This is a free convection problem.
Heat is lost from the door to the surroundings by free convection and radiation.
To determine the rate of heat loss, the door can by modeled as a vertical plate losing
heat by free convection to an ambient air.
As a first approximation, radiation can be neglected.
Newton’s law of cooling gives the rate of heat transfer.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
For laminar flow the solution of Section 7.4 is applicable.
Problem 7.5
This is a free convection and radiation problem.
The geometry is a vertical plate.
Surface temperature is uniform.
Newton’s law of cooling gives convection heat transfer rate while Stefan-Boltzmann law
gives radiation heat transfer rate.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
For laminar flow the solution of Section 7.4 is applicable.
Since radiation heat transfer is considered in this problem, all temperatures should be
expressed
Problem 7.6
This is a free convection problem.
The power dissipated in the electronic package is transferred to the ambient fluid by
free convection.
As the power is increased, surface temperature increases.
The maximum power dissipated corresponds to the maximum allowable surface
temperature.
Surface temperature is related to surface heat transfer by Newton’s law of cooling.
The problem can be modeled as free convection over a vertical plate.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
For laminar flow the solution of Section 7.4 is applicable.
The fluid is air.
Problem 7.7
This is a free convection problem.
The power dissipated in the electronic package is transferred to the ambient fluid by
free convection.
As the power is increased, surface temperature increases.
The maximum power dissipated corresponds to the maximum allowable surface
temperature.
Surface temperature is related to surface heat transfer by Newton’s law of cooling.
The problem can be modeled as free convection over a vertical plate.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
For laminar flow the solution of Section 7.4 is applicable.
The fluid is water.
Problem 7.8
This is a free convection problem.
The surface is maintained at uniform temperature.
Newton’s law of cooling determines the heat transfer rate.
Heat transfer rate depends on the heat transfer coefficient.
The heat transfer coefficient decreases with distance from the leading edge of the plate.
The width of each triangle changes with distance from the leading edge.
For laminar flow the solution of Section 7.4 is applicable.
Problem 7.9
This is a free convection problem over a vertical plate.
The surface is maintained at uniform temperature.
Local heat flux is determined by Newton’s law of cooling.
Heat flux depends on the local heat transfer coefficient
Free convection heat transfer coefficient for a vertical plate decreases with distance from
the leading edge. Thus, the flux also decreases.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent. For
Laminar flow the solution of Section 7.4 is applicable.
The fluid is air.
Problem 7.10
This is a free convection problem over a vertical plate.
The power dissipated in the chips is transferred to the air by free convection.
This problem can be modeled as free convection over a vertical plate with constant
surface heat flux.
Surface temperature increases as the distance from the leading edge is increased. Thus,
the maximum surface temperature occurs at the top end of the plate (trailing end).
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
For laminar flow the analysis of Section 7.5 gives surface temperature distribution.
The fluid is air.
Properties depend on the average surface temperature Ts . Since Ts is unknown, the
problem must be solved by trail and error.
Problem 7.11
This is a free convection problem over a vertical plate.
The power dissipated in the chips is transferred to the air by free convection
This problem can be modeled as free convection over a vertical plate with constant
surface heat flux.
Surface temperature increases as the distance from the leading edge is increased. Thus,
the maximum surface temperature occurs at the top end of the plate (trailing end).
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
For laminar flow the analysis of Section 7.5 gives surface temperature distribution.
The fluid is air.
Properties depend on the average surface temperature Ts . Since Ts is unknown, the
problem must be solved by trail and error.
Problem 7.12
This is a free convection problem over a vertical plate at uniform surface temperature.
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
The integral method can be used to determine the velocity and temperature distribution.
Application of the integral method reduces to determining the velocity and temperature
boundary layer thickness.
Problem 7.13
This is a free convection problem over a vertical plate at uniform surface heat flux.
In general, to determine the Nusselt number it is necessary to determine the velocity and
temperature distribution.
The integral method can be used to determine the velocity and temperature distribution.
Application of the integral method reduces to determining the velocity and temperature
boundary layer thickness.
PROBLEM 8.2
This is an order-of-magnitude, scalar analysis.
PROBLEM 8.5
This problem is restricted to Cartesian coordinates.
PROBLEM 8.6
This problem is restricted to Cartesian coordinates.
PROBLEM 8.7
This problem is restricted to Cartesian coordinates.
PROBLEM 8.8
This problem is restricted to flow over a flat plate, presumably turbulent flow.
PROBLEM 8.10
This analysis applies to turbulent flow over a flat plate.
PROBLEM 8.11
This problem applies to turbulent flow over a flat plate. The lower limit is 1, instead of zero, to
avoid a singularity when plotting in logarithmic coordinates.
PROBLEM 8.12
This problem is application of the momentum integral method to turbulent flow over a flat plate.
The analysis is essentially the same as the Prandtl-von Kármán solution presented in Section
8.4.3.
PROBLEM 8.13
This problem is application of the momentum integral method to turbulent flow over a flat plate.
The analysis is similar to the Prandtl-von Kármán solution presented in Section 8.4.3.
PROBLEM 8.14
This problem is application of the momentum-heat transfer analogy method to turbulent flow
over a flat plate.
PROBLEM 8.16
This problem is restricted to turbulent flow over a flat plate.
PROBLEM 8.17
This problem is pertains to mixed (laminar and turbulent) flow over a flat plate.
PROBLEM 8.18
This problem is an application of a momentum-heat transfer analogy to flow over a flat plate.
PROBLEM 8.19
This problem applies to mixed (laminar and turbulent) flow over a flat plate.
PROBLEM 9.2
This is a turbulent pipe flow problem involving velocity entry length.
PROBLEM 9.3
This is a turbulent pipe flow problem involving velocity entry length.
PROBLEM 9.4
This is a derivation in cylindrical coordinates.
PROBLEM 9.5
This derivation is similar to that of Problem 9.4, where the goal was to evaluating the mean
velocity of a flow with a 1/n power law velocity profile.
PROBLEM 9.7
This is an application of the universal velocity profile.
PROBLEM 9.8
This is an application of the momentum equation to pipe flow. The object of this problem is to
relate the friction factors that we predict in this text to the concept of head loss.
PROBLEM 9.9
This is an application of various friction factor models for smooth circular pipe.
PROBLEM 9.10
This is an application of turbulent, internal forced convection in a circular tube. The problem is
nearly identical to Problem 6.16, except the velocity of the flow is higher, presumably turbulent.
PROBLEM 9.11
This is an application of turbulent heat transfer modeling to a smooth circular pipe.
PROBLEM 9.12
This is an application of turbulent heat transfer modeling to a smooth circular pipe.
PROBLEM 9.13
This is an application of turbulent heat transfer modeling to a smooth circular pipe.
This problem is identical to Problem 9.12, except that the pipe is rough.
PROBLEM 9.14
This is an application of turbulent heat transfer modeling to a smooth circular pipe.
This problem involves cast iron pipe, which implies that we should take roughness into accout.
PROBLEM 9.15
This is an analysis comparing the relative effects of laminar and turbulent flow in a smooth
circular pipe.
PROBLEM 9.16
This is an application of turbulent heat transfer analysis in a rough, non-circular duct.
Problem 10.1
This is an external forced convection problem.
The geometry can be modeled as a flat plate.
Surface temperature is uniform.
Newton’s law of cooling gives heat transfer rate from the surface to the air.
The average heat transfer coefficient must be determined.
The Reynolds number should be evaluated to establish if the flow is laminar, turbulent or
mixed.
Analytic or correlation equations give the heat transfer coefficient.
Problem 10.2
This is an external forced convection problem.
The geometry can be modeled as a flat plate.
Surface temperature is uniform.
To determine the heat flux at a given location, the local heat transfer coefficient must be
determined.
The average heat transfer coefficient is needed to determine the total heat transfer rate.
Newton’s law of cooling gives surface flux and total heat transfer rate.
The Reynolds number should be checked to establish if the flow is laminar, turbulent or
mixed.
Analytic or correlation equations give the heat transfer coefficient.
Problem 10.3
This is an external forced convection problem of flow over a flat plate.
Surface temperature is assumed uniform.
The heat transfer coefficient in turbulent flow is greater than that in laminar flow.
Thus higher heat transfer rates can be sustained in turbulent flow than laminar flow.
The Reynolds number should be checked to establish if the flow is laminar, turbulent or
mixed.
Heat loss from the surface is approximately equal to the power dissipated in the package.
Newton’s law of cooling gives a relationship between heat transfer rate, surface area, heat
transfer coefficient, surface temperature and ambient temperature.
The fluid is air.
Problem 10.4
This is an external forced convection problem.
The geometry is a flat plate.
Surface temperature is uniform.
Newton’s law of cooling gives the heat transfer rate.
The Reynolds number should be checked to establish if the flow is laminar, turbulent or
mixed.
Analytic or correlation equations give the heat transfer coefficient.
If the flow is laminar throughout, heat transfer from the first half should be greater than
that from the second half.
Second half heat transfer can be obtained by subtracting first half heat rate from the heat
transfer from the entire plate.
The fluid is water.
Problem 10.5
The chip is cooled by forced convection.
This problem can be modeled as a flat plate with an unheated leading section.
Newton's law of cooling can be applied to determine the rate of heat transfer between the
chip and the air.
Check the Reynolds number to establish if the flow is laminar or turbulent.
Problem 10.6
Heat transfer from the collector to the air is by forced convection.
This problem can be modeled as a flat plate with an unheated leading section.
Newton's law of cooling can be applied to determine the rate of heat transfer between the
collector and air.
The heat transfer coefficient varies along the collector.
The Reynolds number should be computed to establish if the flow is laminar or turbulent.
Problem 10.7
This is an external forced convection problem.
The flow is over a flat plate.
Surface temperature is uniform.
Plate orientation is important.
Variation of the heat transfer coefficient along the plate affects the total heat transfer.
The heat transfer coefficient for laminar flow decreases as the distance from the leading
edge is increased. However, at the transition point it increases and then decreases again.
Higher rate of heat transfer may be obtained if the wide side of a plate faces the flow. On
the other hand, higher rate may be obtained if the long side of the plate is in line with the
flow direction when transition takes place
The fluid is water.
Problem 10.8
This is an external forced convection problem.
The flow is over a flat plate.
The problem can be modeled as flow over a flat plate with uniform surface heat flux.
Surface temperature varies with distance along plate. The highest surface temperature is
at the trailing end.
Tripping the boundary layer at the leading edge changes the flow from laminar to
turbulent. This increases the heat transfer coefficient and lowers surface temperature.
Newton’s law of cooling gives surface temperature.
Problem 10.9
This is an external forced convection problem.
The flow is normal to a tube.
Surface temperature is uniform.
Tube length is unknown.
Newton’s law of cooling can be used to determine surface area. Tube length is related to
surface area.
The fluid is water.
Problem 10.10
Heat is removed by the water from the steam causing it to condense.
The rate at which steam condenses inside the tube depends on the rate at which heat is
removed from the outside surface.
Heat is removed from the outside surface by forced convection.
This is an external forced convection problem of flow normal to a tube. (v) Newton’s law
of cooling gives the rate of heat loss from the surface.
Problem 10.11
Electric power is dissipated into heat and is removed by the water.
This velocity measuring concept is based on the fact that forced convection heat transfer
is affected by fluid velocity.
Velocity affects the heat transfer coefficient which in term affects surface temperature.
Newton’s law of cooling relates surface heat loss to the heat transfer coefficient, surface
area and surface temperature.
This problem can be modeled as external flow normal to a cylinder.
The fluid is water.
Problem 10.12
This is an external forced convection problem.
The flow is normal to a rod.
Surface heat transfer rate per unit length is known. However, surface temperature is
unknown.
In general, surface temperature varies along the circumference. However, the rod can be
assumed to have a uniform surface temperature.
This problem can be modeled as forced convection normal to a rod with uniform surface
flux or temperature.
Newton’s law of cooling gives surface temperature.
The fluid is air.
Problem 10.13
Electric power is dissipated into heat and is removed by the fluid.
This velocity measuring instrument is based on the fact that forced convection heat
transfer is affected by fluid velocity.
velocity affects the heat transfer coefficient which in term affects surface temperature and
heat flux.
Newton’s law of cooling relates surface heat loss to the heat transfer coefficient, surface
area and surface temperature.
This problem can be modeled as external flow normal to a cylinder.
The fluid is air.
Problem 10.14
The sphere cools off as it drops. Heat loss from the sphere is by forced convection.
The height of the building can be determined if the time it takes the sphere to land is
known.
Time to land is the same as cooling time.
Transient conduction determines cooling time.
If the Biot number is less than 0.1, lumped capacity method can be used to determine
transient temperature.
Cooling rate depends on the heat transfer coefficient.
Problem 10.15
The electric energy dissipated inside the sphere is removed from the surface as heat by
forced convection.
This problem can be modeled as external flow over a sphere.
Newton’s law of cooling relates heat loss from the surface to heat transfer coefficient,
surface area and surface temperature. (iv) The fluid is air.
Problem 10.16
The sphere cools off as it drops. Heat loss from the sphere is by forced convection.
This is an external flow problem with a free stream velocity that changes with time.
This is a transient conduction problem. The cooling time is equal to the time it takes the
sphere to drop to street level.
If the Biot number is less than 0.1, lumped capacity method can be used to determine
transient temperature.
Cooling rate depends on the heat transfer coefficient.
Problem 10.17
This is an internal forced convection problem.
The channel is a tube.
The outside surface is maintained at a uniform temperature.
Neglecting tube thickness resistance means that the inside and outside surface
temperatures are identical.
Fluid temperature is developing.
Inlet and outlet temperatures are known.
The Reynolds number should be determined to establish if the flow is laminar or
turbulent.
The required tube length depends on the heat transfer coefficient.
The fluid is water.
Problem 10.18
This is an internal force convection problem.
The channel is a tube.
The surface is maintained at a uniform temperature.
The velocity is fully developed.
The temperature is developing.
The outlet temperature is unknown.
The Reynolds number should be checked to establish if the flow is laminar or turbulent.
The fluid is air.
Problem 10.19
This is an internal forced convection problem
Tube surface is maintained at uniform temperature.
The velocity is fully developed.
The length of tube is unknown.
The temperature is developing. However, depending on tube length relative to the
thermal entrance length, temperature may be considered fully developed throughout.
The Reynolds number should be checked to determine if the flow is laminar or turbulent.
The fluid is water.
Problem 10.20
This is an internal forced convection problem.
Tube surface is maintained at a uniform temperature.
The velocity and temperature are developing. Thus, entrance effects may be important.
The outlet temperature is unknown.
The fluid is air.
Problem 10.21
This is an internal forced convection problem.
Tube surface is maintained at uniform temperature.
The section of interest is far away from the inlet. This means that flow and temperature
can be assumed fully developed and the heat transfer coefficient uniform.
It is desired to determine the surface flux at this section. Newton’s law of cooling gives a
relationship between local flux, surface temperature and heat transfer coefficient.
The Reynolds number should be checked to determine if the flow is laminar or turbulent.
The fluid is water.
Problem 10.22
This is an internal forced convection problem in a tube.
Both velocity and temperature are fully developed.
Tube surface is maintained at uniform temperature.
The Reynolds number should be computed to establish if flow is laminar or turbulent.
Mean velocity, mean inlet and outlet temperatures and tube diameter are known.
The fluid is air.
Problem 10.23
This is an internal forced convection problem.
The surface of each tube is maintained at uniform temperature which is the same for both.
The velocity and temperature are fully developed. Thus, the heat transfer coefficient is
uniform.
Air flows through each tube at different rates.
The Reynolds number should be computed to establish if the flow is laminar or turbulent.
Surface heat flux depends on the heat transfer coefficient.
Problem 10.24
This is an internal forced convection problem.
The geometry consists of two concentric tubes.
Air flows in the inner tube while water flows in the annular space between the two tubes.
The Reynolds number should be computed for both fluids to establish if the flow is
laminar or turbulent.
Convection resistance depends on the heat transfer coefficient.
Problem 10.25
This is an internal forced convection problem.
The geometry consists of a tube concentrically placed inside a square duct,.
Water flows in the tube and the duct.
The Reynolds number should be computed for the two fluids to establish if the flow is
laminar or turbulent.
Far away from the inlet the velocity and temperature may be assumed fully developed.
Problem 10.26
Heat is lost from the door to the surroundings by free convection and radiation.
To determine the rate of heat loss, the door can by modeled as a vertical plate losing
heat by free convection to an ambient air and by radiation to a large surroundings.
Newton’s law of cooling gives the rate of heat transfer by convection and StefanBoltzmann relation gives the heat loss by radiation.
Problem 10.27
This is a free convection and radiation problem.
The geometry is a vertical plate.
Surface temperature is uniform.
Newton’s law of cooling gives convection heat transfer rate while Stefan-Boltzmann law
gives radiation heat transfer rate.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
Since radiation heat transfer is considered in this problem, all temperatures should be
expressed in degrees kelvin.
The fluid is air.
Problem 10.28
This is a free convection problem.
The power dissipated in the electronic package is transferred to the ambient fluid by
free convection.
As the power is increased, surface temperature increases.
The maximum power dissipated corresponds to the maximum allowable surface
temperature.
Surface temperature is related to surface heat transfer by Newton’s law of cooling.
The problem can be modeled as free convection over a vertical plate.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
The fluid is air.
Problem 10.29
This is a free convection problem.
The power dissipated in the electronic package is transferred to the ambient fluid by
free convection.
As the power is increased, surface temperature increases.
The maximum power dissipated corresponds to the maximum allowable surface
temperature.
Surface temperature is related to surface heat transfer by Newton’s law of cooling.
The problem can be modeled as free convection over a vertical plate.
The Rayleigh number should be computed to determine if the flow is laminar or
turbulent.
The fluid is water.
Problem 10.30
This is a free convection problem.
The surface is maintained at uniform temperature.
The heat transfer coefficient decreases with distance from the leading edge of the plate.
The heat transfer rate from the lower half 1 is greater than that from the upper half 2.
Total heat transfer from each half can be determined using the average heat transfer
coefficient.
Heat transfer from the upper half is equal to the heat transfer from the entire plate minus
heat transfer from the lower half.
Problem 10.31
This is a free convection problem.
The surface is maintained at uniform temperature.
The heat transfer coefficient decreases with distance from the leading edge of the plate.
The width of each triangle changes with distance from the leading edge.
Problem 10.32
This is a free convection problem over a vertical plate.
The surface is maintained at uniform temperature.
Local heat flux is determined by Newton’s law of cooling.
Heat flux depends on the local heat transfer coefficient.
Free convection heat transfer coefficient for a vertical plate decreases with distance from
the leading edge. Thus, the flux also decreases.
The Rayleigh number should be computed to select an appropriate Nusselt number
correlation equation.
The fluid is air.
Problem 10.33
This is a free convection problem over a vertical plate.
The power dissipated in the chips is transferred to the air by free convection.
This problem can be modeled as free convection over a vertical plate with constant
surface heat flux.
Surface temperature increases as the distance from the leading edge is increased. Thus,
the maximum surface temperature occurs at the top end of the plate (trailing end).
Newton’s law of cooling relates surface temperature to heat flux and heat transfer
coefficient.
The fluid is air.
Problem 10.34
Power supply to the disk is lost from the surface to the surroundings by free convection
and radiation.
To determine the rate of heat loss, the disk can by modeled as a horizontal plate losing
heat by free convection to an ambient air and by radiation to a large surroundings.
Newton’s law of cooling gives the rate of heat transfer by convection and StefanBoltzmann relation gives the heat loss by radiation.
Free convection correlations give the heat transfer coefficient.
Conservation of energy at the surface gives the emissivity, if it is the only unknown.
Problem 10.35
This is a free convection problem.
The geometry is a flat plate.
Heat transfer from two plates is to be compared. One plate is vertical and the other is
inclined. Both plates fit in the same vertical space. Thus, the inclined plate is longer than
the vertical plate.
Both plates are maintained at uniform surface temperature.
Heat transfer depends on surface area and average heat transfer coefficient.
Problem 10.36
This is a free convection problem.
The kiln has four vertical sides and a horizontal top.
All surfaces are at the same uniform temperature.
Newton’s law of cooling gives the heat transfer rate.
The sides can be modeled as vertical plates and the top as a horizontal plate.
The fluid is air.
Problem 10.37
Heat transfer from the surface is by free convection and radiation.
The burner can be modeled as a horizontal disk with its heated side facing down.
Newton’s law of cooling gives heat transfer by convection and Stefan-Boltzmann
relations gives heat transfer by radiation.
Both convection and radiation depend on surface temperature.
If the burner is well insulated at the bottom heated surface and its rim, then the electric
power supply is equal to surface heat transfer.
Problem 10.38
Heat transfer from the surface is by free convection and radiation.
The sample can be modeled as a horizontal disk with its heated side facing down or up.
Newton’s law of cooling gives heat transfer by convection and Stefan-Boltzmann relation
gives heat transfer by radiation.
Radiation depends on surface emissivity.
If the disk is well insulated at the heated surface and its rim, then the electric power
supply is equal to surface heat transfer.
Since the electric power is the same for both orientations, it follows that surface heat
transfer rate is also the same.
Each orientation has its own Nusselt number correlation equation.
Problem 10.39
This is a free convection problem.
Heat is transferred from the cylindrical surface and top surface of tank to the ambient air.
Under certain conditions a vertical cylindrical surface can be modeled as a vertical plate.
Newton’s law of cooling gives the heat transfer rate from tank.
The fluid is air.
Problem 10.40
This is a free convection problem.
The geometry is a horizontal round duct.
Heat is transferred from duct surface to the ambient air.
According to Newton’s law of cooling, the rate of heat transfer depends on the heat
transfer coefficient, surface area and surface and ambient temperatures.
Problem 10.41
This is a free convection problem.
The geometry is a horizontal pipe.
Heat is transferred from pipe surface to the ambient air.
Adding insulation material reduces heat loss from pipe.
According to Newton’s law of cooling, the rate of heat transfer depends on the heat
transfer coefficient, surface area and surface and ambient temperatures
Heat transfer coefficient and surface area change when insulation is added.
The fluid is air.
Problem 10.42
This is a free convection problem.
The geometry is a horizontal wire (cylinder).
Under steady state conditions the power dissipated in the wire is transferred to the
surrounding air.
According to Newton’s law of cooling, surface temperature is determined by the heat
transfer rate, heat transfer coefficient, surface area and ambient temperature.
The fluid is air.
Problem 10.43
This is a free convection problem.
The geometry is a horizontal tube.
Under steady state conditions the power dissipated in the neon tube is transferred to the
surrounding air.
According to Newton’s law of cooling, surface temperature is determined by the heat
transfer rate, heat transfer coefficient, surface area and ambient temperature.
The fluid is air.
Problem 10.44
This is a free convection problem.
The geometry is a round horizontal round duct.
Heat is transferred from the ambient air to the duct.
According to Newton’s law of cooling, the rate of heat transfer to the surface depends on
the heat transfer coefficient, surface area and surface and ambient temperatures.
The fluid is air.
Problem 10.45
This is a free convection and radiation problem
The geometry is a sphere.
Under steady state conditions the power dissipated in the bulb is transferred to the
surroundings by free convection and radiation and through the base by conduction.
According to Newton’s law of cooling and Stefan-Boltzmann radiation law, heat loss
from the surface depends on surface temperature.
The ambient fluid is air.
Problem 10.46
At steady state, power supply to the sphere must be equal to the heat loss from the surface
Heat loss from the surface is by free convection.
The surface is maintained at uniform temperature.
Problem 10.47
Heat is transferred from the ambient air to the water in the fish tank.
Adding an air enclosure reduces the rate of heat transfer.
To estimate the reduction in cooling load, heat transfer from the ambient air to the water
with and without the enclosure must be determined.
) Neglecting the thermal resistance of glass, the resistance to heat transfer form the air to
the water is primarily due to the air side free convection heat transfer coefficient.
Installing an air cavity introduces an added thermal resistance.
The problem can be modeled as a vertical plate and as a vertical rectangular enclosure.
The outside surface temperature of the enclosure is unknown.
Newton’s law of cooling gives the heat transfer rate.
The Rayleigh number should be determined for both vertical plate and rectangular
enclosure so that appropriate correlation equations for the Nusselt number are selected.
However, since the outside surface temperature of the enclosure is unknown, the
Rayleigh number can not be determined. The problem must be solved using an iterative
procedure.
Problem 10.48
Heat is transferred from the inside to the outside.
Adding i an air enclosure reduces the rate of heat transfer.
To estimate the savings in energy, heat transfer through the single and double pane
windows must be determined.
The double pane window introduces an added glass conduction resistance and a cavity
convection resistance.
the problem can be modeled as a vertical rectangular enclosure.
Newton’s law of cooling gives the heat transfer rate
The aspect ratio and Rayleigh number should be determined for the rectangular enclosure
so that an appropriate correlation equation for the Nusselt number can be selected.
Problem 10.49
Heat is transferred through the door from the inside to the outside.
Newton’s law of cooling gives the heat transfer rate.
The aspect ratio and Rayleigh number should be determined for the rectangular enclosure
so that an appropriate correlation equation for the Nusselt number can be selected.
The baffle divides the vertical cavity
Problem 10.50
Heat is transferred through the skylight from the inside to the outside.
Newton’s law of cooling gives the heat transfer rate.
The aspect ratio and Rayleigh number should be determined for the rectangular enclosure
so that an appropriate correlation equation for the Nusselt number can be selected.
Problem 10.51
Power requirement is equal to the heat transfer rate through the enclosure.
The problem can be modeled as a rectangular cavity at specified hot and cold surface
temperatures.
The inclination angle varies from 0 o to 175o .
Newton’s law of cooling gives the heat transfer rate.
The aspect ratio and critical inclination angle should be computed to determine the
applicable correlation equation for the Nusselt number.
Problem 10.52
The absorber plate is at a higher temperature than the ambient air. Thus heat is lost
through the rectangular cavity to the atmosphere
The problem can be modeled as an inclined rectangular cavity at specified hot and cold
surface temperatures.
Newton’s law of cooling gives the heat transfer rate.
The aspect ratio and critical inclination angle should be computed to determine the
applicable correlation equation for the Nusselt number.
Problem 8.53
Heat is transferred through the annular space from the outer cylinder to the inner.
Newton’s law of cooling gives the heat transfer rate.
The Rayleigh number should be determined for the enclosure formed by the concentric
cylinders so that an appropriate correlation equation can be selected.
The cylinders are horizontally oriented.
Problem 11.1
Definitions of Knudsen number, Reynolds number, and Mach number are needed.
Fluid velocity appears in the definition of Reynolds number and Mach number.
Problem 11.2
The definition of friction factor shows that it depends on pressure drop, diameter, length
and mean velocity.
Mean velocity is determined from flow rate measurements and channel flow area.
Problem 11.3
The determination of the Nusselt number requires the determination of the temperature
distribution.
Temperature field depends on the velocity field.
The velocity field for Couette flow with a moving upper plate is give in Section 11.6.2.
The solution to the energy equation gives the temperature distribution.
Problem 11.4
Temperature distribution depends on the velocity field.
The velocity field for Couette flow with a moving upper plate is give in Section 11.6.2.
The solution to the energy equation gives the temperature distribution.
Two temperature boundary conditions must be specified.
Temperature distribution and Fourier’s law give surface heat flux.
Problem 11.5
To determine mass flow rate it is necessary to determine the velocity distribution.
Velocity slip takes place at both boundaries of the flow channel.
Because plates move in opposite directions, the fluid moves in both directions. This
makes it possible for the net flow rate to be zero.
Problem 11.6
Model channel flow as Couette flow between parallel plates.
Apply Fourier’s law at the housing surface to determine heat leaving the channel.
Apply the Navier-Stokes equations and formulate the velocity slip boundary conditions.
Follow the analysis of Section 11.6.2 and Example 11.1.
Use the energy equation to determine the temperature distribution
Problem 11.7
To determine the temperature of the lower plate, fluid temperature distribution must be
known.
Temperature distribution depends on the velocity field.
The velocity field for Couette flow with a moving upper plate is given in Section 11.6.2.
The solution to the energy equation gives the temperature distribution.
Two temperature boundary conditions must be specified.
Problem 11.8
To use the proposed approach, the solution to the axial velocity distribution must be
know.
The velocity distribution for Poiseuille flow between parallel plates is given by equation
(11.30) of Section 11.6.3.
Problem 11.9
This is a pressure driven microchannel Poiseuille flow between parallel plates.
The solution to mass flow rate through microchannels is given in Section 11.6.3.
Channel height affects the Knudsen number.
Problem 11.10
This is a pressure driven microchannel Poiseuille flow.
Since channel height is much smaller than channel width, the rectangular channel can be
modeled as Poiseuille flow between parallel plates.
Channel surface is heated with uniform flux.
The solution to mass flow rate, temperature distribution, and Nusselt number for fully
developed Poiseuille channel flow with uniform surface flux is presented in Section
11.6.3.
Problem 11.11
The problem can be modeled as pressure driven Poiseuille flow between two parallel
plates with uniform surface flux.
Assuming fully developed velocity and temperature, the analysis of Section 11.6.3 gives
the mass flow rate and Nusselt number.
The Nusselt number depends on the Knudsen number, Kn. Since Kn varies along the
channel due to pressure variation, it follows that pressure distribution along the channel
must be determined.
Problem 11.12
This is a pressure driven microchannel Poiseuille flow.
Since channel height is much smaller than channel width, the rectangular channel can be
modeled as Poiseuille flow between parallel plates.
Channel surface is maintained at uniform temperature.
The solution to velocity, pressure, and mass flow rate is presented in Section 11.63.
The solution to the temperature distribution and Nusselt number for fully developed
Poiseuille channel flow with uniform surface temperature is presented in Section 11.6.4.
Surface heat flux is determined using Newton’s law.
Problem 11.13
Cylindrical coordinates should be used to solve this problem.
The axial component of the Navier-Stokes equations must be solved to determine the
axial velocity v z .
The procedure and simplifying assumptions used in the solution of the corresponding
Couette flow between parallel plates, detailed in Section 11.6.2, can be applied to this
case.
Problem 11.14
This a pressure driven Poiseuille flow through a microtube.
The procedure for determining the radial velocity component and axial pressure
distribution is identical to that for slip Poiseuille flow between parallel plates.
The solution to the axial velocity is given by equation (11.74).
Continuity equation gives the radial velocity component.
Axial pressure is determined by setting the radial velocity component equal to zero at the
surface.
Cylindrical coordinates should be used to solve this problem.
Problem 11.15
Cylindrical coordinates should be used to solve this problem.
Axial velocity component is needed to determine mass flow rate.
Equation (11.74) gives the axial velocity for this case.
Since axial velocity vary with radial distance, mass flow rate requires integration of the
axial velocity over the flow cross section area.
The procedure and simplifying assumptions used in the solution of the corresponding
Couette flow between parallel plates, detailed in Section 11.6.2, can be applied to this
problem.
Problem 11.16
To use the proposed approach, the solution to the axial velocity distribution must be
known.
The velocity distribution for Poiseuille flow through tubes is given by equation (11.74) of
Section 11.6.5.
Cylindrical coordinates should be used to solve this problem.
Problem 11.17
This is a pressure driven Poiseuille flow through a microtube.
Tube surface is heated with uniform flux.
The solution to mass flow rate, temperature distribution and Nusselt number for fully
developed Poiseuille flow through a tube with uniform surface flux is presented in
Section 11.6.5.
Problem 11.18
The problem is a pressure driven Poiseuille flow through microtube with uniform surface
heat flux.
The Nusselt number depends on the Knudsen number, Kn. Since Kn varies along the tube
due to pressure variation, it follows that pressure distribution along the tube must be
determined.
Assuming fully developed velocity and temperature, the analysis of Section 11.6.5 gives
axial pressure and Nusselt number variation along tube
The definition of Nusselt number gives the heat transfer coefficient.
Problem 11.19
This is a pressure driven Poiseuille flow through a tube at uniform surface temperature.
Since the flow field is assumed independent of temperature, it follows that the velocity,
mass flow rate and pressure distribution for tubes at uniform surface flux, presented in
Section 11.6.6, are applicable to tubes at uniform surface temperature..
The heat transfer coefficient can be determined if the Nusselt number is known.
The variation of the Nusselt number with Knudsen number for air is shown in Fig. 11.16.
The determination of Knudsen number at the inlet and outlet and Fig. 11.16 establish the
Nusselt number at these locations.
The use of Fig. 11.16 requires the determination of the Peclet number.
Mean temperature variation along the tube is given by equation (6.13). Application of
this equation requires the determination of the average heat transfer coefficient.
Problem 11.20
The problem is a pressure driven Poiseuille flow through microtube at uniform surface
temperature.
The Nusselt number depends on the Knudsen number, Kn. Since Kn varies along the tube
due to pressure variation, it follows that pressure distribution must be determined.
Assuming fully developed velocity and temperature, the analysis of Section 11.6.6 gives
axial pressure and Nusselt number variation along the tube.
The definition of Nusselt number gives the heat transfer coefficient.
The variation of the Nusselt number with Knudsen number and Peclet number for air is
shown in Fig. 11.16.
Mean temperature variation along the tube is given by equation (6.13). Application of
this equation requires the determination of the average heat transfer coefficient.