Heat Convection
Alireza Mashayekh
Fall 2023
Integral Solution
● Integral solution can be
used to determine the
actual y variation of
features such as local
heat flux (𝑞″), thermal
boundary layer thickness
(𝛿𝑇 ), and wall jet
velocity profiles
● So far, we only know the
order of magnitude of
relevant flow and heat
transfer parameters ↓
Integral Solution
● Integrating the momentum
equation (4.17) and the
energy equation (4.8)
from the wall (𝑥 = 0) to a
far enough plane 𝑥 = 𝑋 in
the motionless isothermal
cold reservoir, we obtain
the integral boundary
layer equations for
momentum and energy
Integral Solution
● The length scales of
Table 4.1 are very useful
in selecting the proper
shapes of 𝑣 and 𝑇
profiles to be
substituted into the
integral equations
● We must carry out the integral
●
analysis in two parts, for
Pr > 1 and Pr < 1, as the
boundary layer constitution
changes dramatically across
Pr ∼ 1
The other lesson learned from
scale analysis is that the
velocity profile shape is
governed by two length scales:
one for the wall shear layer,
and another for the overall
thickness of the moving layer
of fluid
Integral Solution
High-Pr Fluids
● A suitable set of
profiles for Pr > 1 fluids
compatible with Fig. 4.4a
is:
● Substituting these
profiles in the integral
equations, and setting
𝑋 → ∞ yields:
● where 𝑉, 𝛿𝑇 , and 𝛿 are
unknown functions of
altitude (𝑦), and
Δ𝑇 = 𝑇0 − 𝑇∞ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
● where 𝑞 is the Pr
function:
Integral Solution
High-Pr Fluids
● So, at this point, we have
two (2) equations, and
three (3) unknowns: 𝑣 𝑦 ,
𝛿 𝑦 , 𝑞(Pr)
● The third equation,
necessary for determining
𝑉, 𝛿, and 𝑞 uniquely, is a
challenging proposition
● Historically, to avoid
this problem, some assumed
that 𝛿 = 𝛿𝑇 or 𝑞 = 1
● However, since a great
deal of the information
relating to boundary
layer geometry is buried
in the 𝛿/𝛿𝑇 function, it
is instructive to do an
integral analysis with
𝛿 ≠ 𝛿𝑇
● It is up to the
researcher to come up
with a third equation
Integral Solution
High-Pr Fluids
● First, we must keep in
mind that integral form
equations are approximate
substitutes for the real
equations to be satisfied
● So we have the freedom to
bring into the analysis
any other condition
(equation) that accounts
approximately for
conservation of momentum
or conservation of energy
● Since the energy equation is,
in a scaling sense, less
ambiguous than the momentum
equation*, it makes sense to
select as a third equation a
force balance: One that is
both clear and analytically
brief is the statement that in
the no-slip layer 0 < 𝑥 < 0+ ,
the inertia terms of momentum
equation are zero:
* Because in natural boundary layer flow, the energy equation spells 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 ∼ 𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛, whereas
the momentum equation spells either 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 ∼ 𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦 or 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 ∼ 𝑏𝑢𝑜𝑦𝑎𝑛𝑐𝑦
Integral Solution
High-Pr Fluids
● This is to say that right
next to the wall, there
is no ambiguity
associated with whether
inertia is negligible
compared with both
friction and buoyancy,
regardless of the Prandtl
number
● So, considering 𝛿~𝑦 1/4 and
𝑉~𝑦 1/2 , the following
three equations should be
solved for 𝑉, 𝛿, 𝑞
Integral Solution
High-Pr Fluids
● The main results are the
function 𝑞 Pr :
● In the limit Pr → ∞, the
results reduce to:
● And the local Nusselt
number:
● Which confirm the scaling
laws listed in Table 4.1
Integral Solution
Low-Pr Fluids
● for a Pr < 1 fluid, we
combine the previously
assumed temperature
profile with a new
velocity profile:
● noticing that 𝛿𝑇 ∼ 𝑦 1/4 ,
𝛿𝑣 ∼ 𝑦 1/4 , and 𝑉1 ∼ 𝑦 1/2
yields:
● In the limit Pr → 0:
● where 𝑉1 , 𝛿𝑇 , and 𝛿𝑣 are
unknown functions of 𝑦
Integral Solution
Low-Pr Fluids
● Once again, this limiting
behavior confirms within
a numerical factor of
order 1 the scaling laws
discovered in the
preceding section
Integral Solution
Low-Pr Fluids
● The integral heat transfer
results are summarized in
Fig. 4.6 next to the
similarity solution that
will be outlined in the
next section
● The Nu expressions (4.49)
and (4.53) match at
Pr = 5/12, where the assumed
velocity profiles are
identical (𝑞 = 1, 𝑞1 = 1).
Integral Solution
Low-Pr Fluids
● It should be noted that the
●
Nusselt number calculations
depend to some extent on the
choice of analytical
expressions for velocity and
temperature profiles
This choice is always a tradeoff between what constitutes a
reasonable profile shape and
the function that leads to the
fewest analytical
complications
●
●
●
In our calculations, the choice
of exponentials in the makeup of
temperature and velocity
profiles led to a relatively
simple analysis
Figure 4.6 also shows the
Nusselt number predicted by
Squire’s integral analysis,
which assumes polynomial
temperature and velocity
profiles with 𝛿𝑇 = 𝛿
Although the 𝛿𝑇 = 𝛿 assumption is
justified only for fluids with
Pr~1, the Squire analysis
predicts the correct Nusselt
number in a wide Pr range
Similarity Solution
● We can think of
temperature and wall jet
profiles whose shape
remains unchanged as both
profiles occupy wider
areas as 𝑦 increases
● From Table 4.1 and the
integral solution, we
know that any length
scale of the boundary
layer region is
proportional to 𝑦 1/4
● The dimensionless
similarity variable 𝜂(𝑥, 𝑦)
can then be constructed as
𝑥 divided by any of the
length scales summarized
in Table 4.1
● selecting the Pr > 1 thermal
boundary layer thickness
−1/4
𝑦Ra𝑦
as the most
appropriate length scale,
the similarity variable
emerges as:
Similarity Solution
● Introducing the
streamfunction 𝑢 = 𝜕𝜓/𝜕𝑦,
𝑣 = −𝜕𝜓/𝜕𝑥 in place of the
continuity equation, the
boundary layer equations
become:
● From the first column of
Table 4.1 we note that,
in general, the
dimensionless temperature
profile will be a
function of both 𝜂(𝑥, 𝑦)
and Pr ; let this unknown
function be 𝜃(𝜂, Pr),
defined as:
Similarity Solution
● For the vertical velocity
profile 𝑣, from the
fourth column of Table
4.1 where Pr > 1, we
select the expression:
● From the definition
𝑣 = −𝜕𝜓/𝜕𝑥, we conclude
that the streamfunction
expression must be:
1/2
● where 𝛼/𝑦 Ra𝑦 represents
the scale of 𝑣, and
𝐺(𝜂, Pr) is the
dimensionless similarity
profile of the wall jet
● where 𝐺 = −𝜕𝐹/𝜕𝜂
Similarity Solution
Energy
equations
● Substituting
Momentum
equation
●
●
● In the boundary layer
equations for energy and
momentum, we get this
system of dimensionless
equations:
●
where (·)′ is shorthand notation
for 𝜕 · /𝜕𝜂
These equations show once again
the meaning of the Pr > 1 scaling
adopted in the definition of 𝜂
and 𝐺 (both of order 1)
The energy equation is a balance
between convection and
conduction, while the momentum
equation reduces to a balance
between friction and buoyancy as
Pr → ∞, that is, as the inertia
effect vanishes
Similarity Solution
● These equations must be
solved subject to the
similarity formulation of
the appropriate boundary
conditions:
Similarity Solution
● Figures 4.7a and b
present the solution as
temperature profiles and
velocity profiles in the
thermal boundary layer
region 𝜂 = 𝑂(1)
● in the limit Pr → ∞, the
temperature profiles
collapse onto a single
curve
Similarity Solution
● Also, in the same limit, the
●
𝜂 ∼ 1 portions of the velocity
profiles approach a single
curve, while the dimensionless
velocity peak is consistently
a number of order 1 (the
velocity peak falls in the
region occupied by the thermal
boundary layer)
As Pr increases, the velocity
profile extends farther and
farther into isothermal fluid.
All these observations support
the scale analysis whose
results have been summarized
in Table 4.1
Similarity Solution
● The local heat transfer
coefficient predicted by
the similarity solution
is:
● The numerical coefficient
− 𝜃 ′ 𝜂=0 is, in general, a
function of the Prandtl
number, as shown in Table
4.2 and Fig. 4.6
Similarity Solution
● In the two Pr limits of
interest, the Nusselt
number approaches the
following asymptotes:
● Since ℎ ∼ 𝑦 −1/4 the average
heat transfer coefficient
for a wall of height 𝐻 is
ℎ0−𝐻 = (4/3)ℎ(𝑦 = 𝐻)
● The average Nusselt
number Nu0−𝐻 = ℎ0−𝐻 𝐻/𝑘 is
equal to (4/3)Nu(𝑦 = 𝐻)
● Therefore, the wallaveraged heat transfer
results corresponding to
the two Pr limits are:
Similarity Solution
● These conclusions are
●
anticipated within 30 percent
by the scaling laws of Table
4.1: Such good agreement is
common when the scale analysis
is correct
Figure 4.6 shows that despite
the factor of 10 increase in
the Prandtl number from air
(Pr = 0.72) to water (Pr ≅ 5– 7),
the Nusselt number varies by
only 15 percent if the
Rayleigh number is held
constant
UNIFORM WALL HEAT FLUX
● The analyses presented so
far are based on the
assumption that the
vertical wall is
isothermal
● This would be a good
approximation in cases
where the vertical wall
is massive and highly
conducting in the
vertical 𝑦 direction
● From a practical standpoint,
●
●
however, an equally important
wall model is the uniform heat
flux condition 𝑞″ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
In many applications, the wall
heating effect is the result
of radiation heating from the
other side or, as in the case
of electronic components, the
result of resistive heating
We only outline the scale
analysis for this case
UNIFORM WALL HEAT FLUX
● Regardless of how 𝑞″,
𝑇,
and 𝛿𝑇 vary with altitude
𝑦, the definition of wall
heat flux requires that:
UNIFORM WALL HEAT FLUX
● Figure 4.8a illustrates
this scaling law in the
case of an isothermal
wall, where both Δ𝑇 and
the product 𝑞 ″ 𝛿𝑇 are
independent of 𝑦
● Figure 4.8b shows what to
expect in the case of
constant 𝑞″, namely,
identical Δ𝑇 and 𝛿𝑇 as
functions of 𝑦
UNIFORM WALL HEAT FLUX
● To determine these 𝑦
functions, we make the
observation that the
scaling analysis starting
with eq. (4.19) is
general; in other words,
in that analysis, 𝛿𝑇 and 𝑇
represent the correct
order of magnitudes of
thermal layer thickness
and wall–ambient
temperature difference
along a wall of height 𝐻
● For Pr ≫ 1 fluids, eq.
(4.26) recommends:
● Recognizing that in the
present problem Δ𝑇 is not
given (𝑞″ is), we use
𝑞 ″ ~𝑘Δ𝑇/𝛿𝑇 to eliminate Δ𝑇
and solve for 𝛿𝑇
UNIFORM WALL HEAT FLUX
● Where Ra∗ is a Rayleigh
number based on heat flux
𝑞″:
● The corresponding (Pr ≫ 1)
scale of the wall–ambient
temperature difference
is:
● Note that both 𝛿𝑇 and 𝑇
are proportional to 𝐻1/5
● Because the 𝐻-averaged
quantities are
proportional to 𝐻1/5 , the
local values of 𝛿𝑇 and 𝑇
are proportional to 𝑦 1/5
● The local Nusselt number
for a constant heat flux
wall is defined as:
UNIFORM WALL HEAT FLUX
● Therefore, in the range
Pr ≥ 1, the Nusselt number
must scale as:
● For low-Pr number fluids,
following the same steps:
UNIFORM WALL HEAT FLUX
● The validity of these
scaling results can be
tested by referring to
more exact analyses
published on the same
topic
● Sparrow carried out an
integral analysis of the
same type as Squire’s
(i.e., assuming only one
length scale 𝛿𝑇 for the
velocity profile) and
arrived at the local
Nusselt number:
●
●
The similarity solution was
reported by Sparrow and Gregg,
who found that above equation
is, in fact, an adequate curve
fit for the similarity Nu
results in the range 0.01 < Pr < 100
Thus, in the two Pr limits, the
above equation yields the
following local Nusselt numbers:
UNIFORM WALL HEAT FLUX
● Similarity solutions can
be developed for an
infinity of wall
temperature conditions,
provided that they obey
either:
○
○
○
the power law: 𝑇0 − 𝑇∞ = 𝐴𝑦 𝑚
the exponential law:
𝑇0 − 𝑇∞ = 𝐴𝑒 𝑚𝑦
the line 𝑇0 − 𝑇∞ = 𝐴 + 𝐵𝑦
● where 𝐴, 𝐵, and 𝑚 are all
constants
● Thus, the 𝑇0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 and
𝑞″ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 problems
discussed so far are only
two special cases of the
vast analytically
accessible class of
problems
● From an engineering
standpoint, however, the
𝑇0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 and 𝑞″ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
results are by far the
most useful
Conjugate Boundary Layers
● There are many engineering
situations in which the
vertical wall that heats a
buoyant boundary layer is
itself heated on the back
side by a sinking boundary
layer
○
Such is the case in walls,
partitions, and baffles
encountered regularly in the
thermal design of living
quarters and insulation
systems
Conjugate Boundary Layers
● Boundary layers form on
both sides of the wall;
however, the wall
temperature or heat flux
are not known a priori as
in the simpler models
considered earlier
● The condition of the wall
is the result of the heat
transfer interaction
between the two boundary
layers
● It is said that depending
on the layer-to-layer
interaction, the wall
temperature “floats” to
an equilibrium
distribution between the
two extreme temperatures
maintained by the two
reservoirs
Conjugate Boundary Layers
●
●
●
●
Wall 𝑞″ distribution is
approximated satisfactorily by the
𝑞″ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 model discussed
Figure 4.10 shows the Nusselt
number predicted analytically in
the Pr → ∞ limit based on the Oseenlinearization method
This approach consists of writing
integral conservation equations for
both sides of the wall, with the
additional complication that the
wall temperature 𝑇0 (𝑦) is unknown
The additional equation necessary
for determining 𝑇0 is the condition
of heat flux continuity in the 𝑥
direction, from one face of the
wall to the other
Conjugate Boundary Layers
●
●
●
●
Both the overall Nusselt number
and the Rayleigh number are
based on the overall temperature
difference imposed by the two
fluid reservoirs
The heat transfer rate (hence,
1/4
the ratio Nu0–𝐻 /Ra𝐻 ) decreases
as the wall thickness resistance
parameter ω increases
The dimensionless wall number
proposed is:
where 𝑡, 𝐻, 𝑘, and 𝑘𝑤 are the
wall thickness, wall height,
fluid conductivity, and wall
conductivity, respectively
Vertical Channel Flow
● Consider now the
interaction between the
boundary layers formed
along two parallel walls
facing each other
● If the boundary layer
thickness scales are much
smaller than the wall-towall spacing 𝐷, the flow
along one wall may be
regarded (approximately)
as a wall jet unaffected
by the presence of another
wall
Vertical Channel Flow
● On the other hand, if the
●
boundary layer grows to the
point that its thickness
becomes comparable to 𝐷, the
two wall jets merge into a
single buoyant stream rising
through the chimney formed by
the two walls
It is clear from Fig. 4.12
that the channel flow departs
from the wall jet description
in the same way that the duct
flows of Chapter 3 depart from
the pure boundary layer flows
of Chapter 2
Vertical Channel Flow
● Here we focus on the
simplest analysis of the
channel flow, with the
final objective of
predicting the capability
of this flow to cool or
heat the walls of the
channel
● The flow part of the
problem may be solved by
considering the momentum
equation in the 𝑦
direction:
● The mass continuity equation,
in conjunction with the
assumption that the channel is
long enough so that the u
scale becomes sufficiently
small, leads to the concept of
fully developed flow, for
which we have:
Vertical Channel Flow
● The momentum equation in
the lateral direction 𝑥
can be used to show that
the pressure in the fully
developed region is a
function of 𝑦 only, and,
because both ends of the
channel are open to the
ambient of density 𝜌∞ :
● Combining these equations
and again using the
Boussinesq approximation
yields the much simpler
momentum equation:
● To solve this equation,
it is necessary to derive
the temperature profile 𝑇
− 𝑇∞
Vertical Channel Flow
● The two profiles,
velocity and temperature,
are coupled
● Hence, momentum and the
energy equation must be
solved simultaneously
● A much simpler solution
approach is possible if we
observe that in the fully
developed region between two
isothermal walls, the
temperature difference can be
approximated by 𝑇0 − 𝑇∞ ; in
other words:
● Based on this approximation,
the right-hand side of the
momentum equation becomes a
constant
Vertical Channel Flow
● With this assumption, the
velocity profile will be:
● Mass flow rate per unit
length normal to the
plane:
● Total heat transfer rate
between stream and
channel walls:
● Average heat flux
Vertical Channel Flow
● Overall Nusselt number:
● Note that the
dimensionless group
emerging from this
analysis is the Rayleigh
number based on wall-towall spacing
● and that the Grashof
number is once again
absent from the
discussion
● This conclusion answers
again the question of
whether the Grashof
number is a relevant
dimensionless group in
natural convection: It is
not!
Vertical Channel Flow
● The fully developed flow
and heat transfer
solution is valid for all
Prandtl numbers
● The Rayleigh number range
of its validity follows
from the requirement that
the thermal entrance
length 𝑌𝑇 be much smaller
than the channel height 𝐻
● The order of magnitude of
𝑌𝑇 follows from the
observation that the
thermal boundary layer
thickness 𝛿𝑇 becomes of
order 𝐷/2 when 𝑦 is of
order 𝑌𝑇
Vertical Channel Flow
● Evaluating 𝑌𝑇 from above:
● In conclusion, the fully
developed flow and
temperature profile
assumptions break down if
the Rayleigh number
exceeds an order of
magnitude dictated by the
geometric aspect ratio of
the channel, 𝐻/𝐷
Combined Natural and Forced Convection (Mixed Convection)
● So far, we assumed that
the fluid in the reservoir
is motionless
● However, in practice, in
any room there is an airconditioner that
replenishes the air
continuously or
intermittently
● Which means that air is in
motion:
○
The reservoir is forced into
and out of the room by an
external agent
● Depending on the strength
of this forced
circulation, the heat
transfer from the wall to
the room air may be ruled
by either natural
convection or forced
convection or a
combination of natural
and forced convection
Combined Natural and Forced Convection (Mixed Convection)
● Due to the diversity of
the natural– forced
convection interaction,
it is impossible to treat
this subject fully
● However, it is
instructive to study one
simple configuration and
to experience the power
and cost-effectiveness of
pure scaling arguments
Combined Natural and Forced Convection (Mixed Convection)
● As is shown in Fig. 4.13,
let us consider the heat
transfer from a vertical
heated wall (𝑇0 ) to an
isothermal fluid
reservoir moving upward
(𝑇∞ , 𝑈∞ ), that is, in the
same direction as the
natural wall jet present
when 𝑈∞ = 0
Combined Natural and Forced Convection (Mixed Convection)
● Under what conditions is
the combined natural–
forced phenomenon
characterized
(approximately) by the
scales of pure natural
convection, and
conversely, under what
conditions is it
characterized by the
scales of pure forced
convection?
● If the mechanism is
natural convection, the
thermal distance between
the heat-exchanging
entities is of order:
● On the other hand, if the
mechanism is forced
convection, the wall and
the reservoir are
separated by a thermal
length of order:
Combined Natural and Forced Convection (Mixed Convection)
● The type of convection
mechanism is decided by
the smaller of the two
distances, 𝛿𝑇 𝑁𝐶 or
𝛿𝑇 𝐹𝐶 , because the wall
will leak heat to the
nearest heat sink
● Thus, the scale criterion
for transition from
natural to forced
convection is:
● In other words, for Pr > 1
fluids:
Combined Natural and Forced Convection (Mixed Convection)
● To verify the validity of
this criterion, we examine
the local similarity
solution to the combined
heat transfer problem
● This solution shows that
forced convection
dominates at small values
of the group Gr𝑦 /Re2𝑦 , while
natural convection takes
over at large values of
the same parameter
Combined Natural and Forced Convection (Mixed Convection)
● Note, however, that the
knee in each Nusselt
number curve shifts to
the right as Pr increases
○
This effect is due to the
fact that the abscissa
parameter used, Gr𝑦 /Re2𝑦 , is
not the same as the
dimensionless group that
serves as transition
parameter:
Combined Natural and Forced Convection (Mixed Convection)
● Figure 4.14 shows the
replotting of the
similarity solution using
the transition parameter
on the abscissa and the
forced convection Nusselt
number scaling on the
ordinate
● The sign of correct
scaling is that the Pr > 1
curves fall on top of each
other, and the knee of all
the curves is at 𝑂(1) on
the abscissa
Combined Natural and Forced Convection (Mixed Convection)
● In conclusion, mixed
convection can be
understood and predicted
by intersecting its
asymptotes, natural
convection and forced
convection
● Repeating the geometric
arguments as before, this
time for Pr < 1 fluids, we
find the following
transition criterion: