Experiment in Natural Convection
CE 328 - CHEMICAL ENGINEERING LABORATORY II
Spring 2004
Objective:
The three objectives of this experiment are:
1. To measure temperature profiles in air near a heated vertical plate.
2. To determine the transfer coefficient for natural convection heat transfer from a vertical
plate.
3. To compare the measurements with the theory.
System:
A flat vertical plate made of Lexan is heated on one side by boiling water. Heat is lost to air on
the other side through a natural convection process.
Measurements:
The temperature drop across the Lexan plate and the temperature profiles on the air side are
measured using thermocouples and the data is taken by a LabView program. Bring a floppy disk
for this lab.
Theory:
Natural (or free) convection is observed when density gradients are present in a fluid acted upon
by a gravitational field. Our example of this phenomenon is the heated vertical plate exposed to
air that, far from the plate, is motionless. Near the plate, the air is heated, and since the density
of the heated air is less than that of the ambient air, it rises. An inflow of cooler air replenishes
the rising fluid. A steady state situation is achieved when the temperatures of the plate and the
ambient fluid are held constant. In the ideal conception of the problem, the plate is semi-infinite
in length (where x is measured from the low end of the plate; the other end and lateral edges are
at ) and the air is unbounded. Since the heated surface is stationary, the fluid velocity at the
surface must be zero. Far from the surface, the velocity is also zero. In between, thermal and
momentum boundary layers develop, where the velocity is non-zero. These begin at the bottom
edge of the plate and become thicker with increasing distance up the plate (x). The profiles of the
temperature (T) and the velocity components in the x and y direction (vx and vy) are qualitatively
as shown in figure 1. We can analyze the flow and heat transfer using the mass, momentum and
energy balance equations that govern these processes. These equations are
u  v

0
(1)
x y
u
u   2 u
u


 g x  (T  T )
(2)
x
y  y 2
Vertical Velocity Component (u)
Temperature (T)
(x)
(x)
gx
Tw
u
x
x
v
T
u=v=0
y
y
Figure 1: Qualitative representation of the velocity and temperature fields near the heated
surface. For simplicity, the thermal and momentum boundary layers are assumed to have the
same thickness.
u
T
T
2T
v
 2
x
y
y
(3)
where:
u is the velocity in the vertical (x) direction
v is the velocity in the horizontal (y) direction
x is the distance in the vertical direction measured from the bottom of the plate
y is the distance in the horizontal direction measured from the surface of the plate
 is the dynamic viscosity
 is the fluid density
gx is the acceleration of gravity
 is the volume coefficient of expansion for the fluid ( = 1/T for an ideal gas)
Equation (1) is the mass balance (continuity) equation. Equation (2) is the balance on
momentum in the x direction. The first two terms represent convection of momentum in the x
and y directions. The first term on the right-hand-side is viscous transport of momentum in the y
direction. The last term in equation (2) is the buoyancy force that drives the motion. Equation
(3) is the balance on the fluid enthalpy. The first two terms represent convection of enthalpy in
the x and y directions, respectively. The term on the right-hand-side is transport of enthalpy by
conduction. These equations are simplified forms of the “general” mass, momentum, and
enthalpy balance equations. Many assumptions are implied by our use of the boundary layer
form given above. The physical properties (density, heat capacity, thermal conductivity,
viscosity) are taken to be constant. The buoyance force assumes the Boussinesq approximation,
that the buoyancy driving force can be expressed in terms of a coefficient of expansion and that
density differences can be neglected elsewhere in the equations. Conduction and viscous
transport of momentum in the x direction are neglected as well. For the details of these
approximations and others made in the derivation of the boundary layer equations, as well as the
range of validity of these approximations, see the references given below. A more detailed
derivation of the governing equations is presented in appendix 1.
The boundary conditions for the velocity and temperature fields are:
T = Tw at y = 0
T = T for y  
u = v = 0 at y = 0
u = v = 0 for y  
The equations and boundary conditions given above are those appropriate for a boundary-layer
analysis of the problem. These equations are generally analyzed by assuming that there is a
similarity solution to them. That is, we assume that the shape of the solution is independent of
the position along the plate (x), and that only the boundary-layer thickness changes with x. This
assumption will allow us to convert the partial differential equations into ordinary differential
equations that can be handled more easily. Two solution methods, a differential method and a
more approximate (but simpler) integral method, are outlined in appendices 2 and 3 at the end of
this handout.
The predicted temperature profile that results from these analyses (for laminar flow) can be
approximately represented the similarity equation by
T  T  y 
 1  
To  T  δ 
where  is the boundary layer thickness, given by
2
1
(4)
1
 0.952  Pr 
δ  3.93x 
(5)
 Grx 4 .
2
Pr


In expression (5), Grx is the Grashof number, which is interpreted as the ratio of the buoyancy to
viscous forces. It is defined as
x 3 g x β Tw  T 
(6)
Grx 
2
 μ 
 ρ
and Pr is the Prandtl number, (Pr = /()), which is the ratio of the kinematic viscosity to the
thermal diffusivity.
4
Also of interest is the rate at which heat can be removed from the plate by the air. At the surface
of the plate, in the air stream, the heat flux, qw, is given by:
 dT 

(7)
q w  k 
 dy  y 0
Thus, if the thermal conductivity, k, of the air in contact with the plate is known and if the
temperature gradient at the wall is measured, q w is determined. Conventionally, the heat transfer
characteristics are expressed in terms of the Nusselt number, Nu, which is defined as
xhx
(8)
k
where the local heat transfer coefficient, hx, is defined by "Newton's law of cooling"
q w
(9)
hx 
Tw  T
An approximate analytical expression for Nux can be obtained by combining equations (4), (5),
(7), (8), and (9). The result is:
Nu x 
1
 Pr 2 Grx 

(10)
Nu x  0.508
0
.
952

Pr


An experimental value for hx (and therefore Nux) can be obtained by measuring the temperature
gradient at the wall and determining q w from equation (7), then using this heat flux, along with
measured values of Tw and T, in equation (9).
4
If the plate is sufficiently long, or the temperature difference (Tw - T) is large enough, then the
laminar boundary layer becomes unstable, turbulence develops, and the temperature profiles are
altered. Laboratory measurements show that the flow regimes are approximately delineated by
laminar flow regime
104 < Grx Pr < 109
turbulent flow regime
Grx Pr > 109
In the turbulent flow regime, the equation for the temperature distribution is
T  T
 y
 1  
To  T
δ
1
7
(11)
with
0.1
 1  0.494 Pr 3 

(12)
δ  0.565x Pr 

Grx


These two equations may be combined to obtain an expression equivalent to (but different from)
equation (10). The new Nusselt number correlation becomes
2
5
Gr
8 

x
15
(13)
Nu x  0.0295 Pr 
2 
 1  0.494 Pr 3 
Both equation (10) and equation (13) are generally presented in the literature as a plot of log Nux
vs. log(Grx Pr). In fact, the laminar and turbulent measurements generally appear on a single
graph.
2
8
15
Description of Apparatus
The apparatus is shown schematically in Figure 2; further details are given in Figure 3. The plate (heat
transfer surface) is heated by boiling water circulating in a tank which is insulated on five sides. The
water is heated by means of three electrical heaters that are submerged in the tank. The heat transfer
surface is protected from room air currents by a Plexiglas shield. The locations of the thermocouples are
shown in Figure 3. The function of each of the thermocouples is as follows:
1-6
These thermocouples are used to measure the plate temperature on the air side and to check
for vertical thermal gradients along the plate. These thermocouples are located in the plate
"at" the air-side surface.
7
This thermocouple is used to measure the free stream temperature.
8
This thermocouple is used to measure the fluid temperature profile within the boundary
layer.
The thermocouples are connected to a computerized data acquisition system that is controlled using
LabView. A simple program for data acquisition is provided. The boundary layer probe (thermocouple
8) and the ambient probe (thermocouple 7) are connected directly to the data acquisition board.
Thermocouples 1 through 6 are connected through a switch that allows any one of the thermocouples to
be connected to the data acquisition board. The data acquisition program is described in more detail in
appendix 4. Bring a floppy disk for this lab.
Boiling
Water Tank
Electric
Heaters
Heat Transfer
Surface
Plexiglas
Shield
Boundary
Layer
Probe
(T8)
Ambient
Probe (T7)
Figure 2: Schematic diagram of the apparatus
12”
T6
3”
T5
22”
2”
T4
2”
T3
2”
T2
2”
T1
4”
7”
Figure 3: Front view of the heated plate showing the locations
of the implanted thermocouples
Experimental Data:
Measure T(y) (temperature as a function of distance from the plate surface) for at least 4 values
of x (vertical height) along the plate. Make sure that, in the region immediately adjacent to the
plate, the data points are spaced closely enough to allow an accurate determination of dT dy  y  0 .
To be certain, you may want to plot the data as you collect it. Be sure to record the ambient and
wall temperatures along with the temperature profiles.
Computations:
1. Plot T vs. y for each value of x on a single graph.
2. Determine
dT dy  y0
for each profile. You will probably want to do this by fitting a
function (linear is preferred) through the data points nearest the wall and extrapolating
the function to y = 0. Estimate the uncertainty in this value as well.
3. Calculate hx for each profile using equations (7) and (9). You can also calculate the heat
flux based on the temperature drop across the Lexan plate (and its thermal conductivity
and thickness). Calculate the flux and heat transfer coefficient by this means as well.
4. Plot the T(y) data in the similarity form suggested by equation (4) if laminar, and by
equation (11) if turbulent.
5. Plot the Nux data (Equation (8)) according to equation (10) or (13) (whichever is
appropriate). Remember x is vertical distance here. This means Nux is calculated from
your data, and represented as points on your graph. (You used your experimental data to
calculate this.) Then use Nux calculated by the theory, equations (10) or (13), as a line on
the same graph.
Questions for Pre-Lab:
1. Write a detailed operating procedure.
2. What does a similarity graph represent? What does a similarity graph look like? (When
you plot it in similarity form, what are the axes, what shape and slope do you expect?)
3. What distances from the wall do you plan to use?
4. What points (distance from the wall) do you plan to use to determine
dT dy  y 0 ?
5. Do you understand how to use the micrometer?
Questions to be addressed in writing the report:
1. Is there "similarity" for the temperature profiles? If not, why?
2. How does your similarity curve compare with others reported in the literature?
3. How do your results for the Nusselt number and its dependence on Grx compare to those
reported in the literature? (Discuss your results from question 5, above.)
4. Is the flow laminar or turbulent? How do you know?
5. Compare the values of hx computed using
dT dy  y0
to those based on the temperature
drop across the Lexan plate. How do they compare with values reported in the
literature?
6. Do the boundary conditions in the experiment conform to those in the theory? If there are
differences, what are they?
References:
1. Holman, J.P.; Heat Transfer; McGraw-Hill Book Co., Inc., New York (1963).
2. Hartnett, J.P.; Recent advances in Heat and Mass Transfer; McGraw-Hill Book Co., New
York (1961).
3. Eckert, E.R.G..and Drake, R.M.; Heat and Mass Transfer; McGraw-Hill Book Co., New
York (1959).
4. Bird, R.B., Stewart, W.E. and Lightfoot, E.N.; Transport Phenomena, John Wiley and
Sons, Inc.; New York (1960).
5. Welty, J.R., Wicks, C.E. and Wilson, R.E.; Fundamentals of Momentum, Heat and Mass
Transfer; John Wiley and Sons, New York (1976).
6. Parker, J.D., Boggs, J.H. and Blick E.F.; Introduction to Fluid Mechanics and Heat
Transfer; Addison-Wesley Publishing Co., Reading Mass. (1969).
7. Rohsenow, W.M. and Choi, H.Y.; Heat, Mass and Momentum Transfer; Prentice-Hall
(1961).
8. Kays, W.M. and Crawford, M.E.; Convective Heat and Mass Transfer; McGraw-Hill, Inc.
(1993).
Appendix 1: Derivation of governing equations for free-convection boundary
layer.
Key assumptions:
(1) Constant properties (, , k, Cp), except for the variation in density that drives the flow.
(2) Pressure gradients perpendicular to the plate can be neglected.
(3) Density variation can be approximated by a linear dependence on temperature. This is called
the Boussinesq approximation (see below).
(4) Diffusive transport (of both momentum and energy) in the direction parallel to the plate can
be neglected.
There are also, of course, many other implied assumptions (steady-state situation, viscous
dissipation is negligible in the energy equation, everything is constant in the z-direction (parallel
to plate, perpendicular to gravity).
First, we will look up the “general” governing equations in a reference text. In Bird, Stewart, and
Lightfoot, we find the following equations. We can neglect any terms involving the z coordinate
or the corresponding velocity component, as well as the transient terms. This leaves:
Continuity equation (overall mass balance)

 u    (  )  0
x
y
Momentum balance in x direction (parallel to plate and gravity) for a Newtonian fluid with
constant , and .
  2u  2u 
 u
u 
P


  u         2  2   g x
y 
x
y 
 x
 x
Energy balance for a Newtonian fluid with constant , and .
  2T  2T 
 T
T 
  k  2  2 
C p  u

y 
y 
 x
 x
We have already used approximation (1) in choosing which version of the equations to copy
from the text. In approximation (2), we say that there are no pressure gradients perpendicular to
P
the plate. This means that the axial pressure gradient,
, is independent of y. Far from the
x
plate, where the velocity is zero, this pressure gradient is just the hydrostatic pressure gradient:
P
   gx
x
Substituting this hydrostatic pressure gradient into the momentum balance gives
  2u  2u 
 u
u 
  u       2  2       g x
y 
y 
 x
 x
Now, we apply assumption (3), the Boussinesq approximation. This says that we can write the
density variation as
       (T  T )
where  is the coefficient of thermal expansion of the fluid, defined as
1   
   
  T  T T
For an ideal gas, this is
RT    Mp  
1
     

 
Mp  T  RT   T T
T

Applying the Boussinesq approximation to the momentum equation gives
  2u  2u 
 u
u 
  u       2  2   g x  (T  T )
y 
y 
 x
 x
Finally, according to approximation (4), we can neglect the diffusive terms in the x-direction.
Doing this, dividing the continuity and momentum equations through by the density, and
dividing the energy equation through by the density and specific heat gives us the governing
equations that we will use to analyze this situation.
 u v

0
x y
u
u   2 u
u


 g x T  T 
x
y  y 2
u
T
T
2T
v
 2
x
y
y
Appendix 2: Approximate Integral Method of Analyzing the Free Convection
Boundary Layer Equations
The governing equations for the free convection boundary layer on a vertical plate are:
u  v

0
(1)
x y
u
u   2 u
(2)
u


 g x T  T 
x
y  y 2
T
T
2T
(3)
u
v
 2
x
y
y
with boundary conditions:
T = Tw at y = 0
T = T for y  
u = v = 0 at y = 0
u = v = 0 for y  
These can be formally integrated in the y direction from y = 0 to y = Y. First, it is convenient to
re-write the left-hand-sides of equations (2) and (3) as
u
u  2

  2u
(4)
u


u  u  
 g x T  T 
x
y x
y
 y 2
 
T
T


 2T
(5)

 u T  T    T  T    2 T  T 
x
y x
y
y
Equation (1) is used to go from the form of equations (2) and (3) to that of equations (4) and (5).
It is also more convenient to work everywhere in terms of the temperature difference from the
temperature far from the plate rather than the absolute temperature. Formally integrating these
from y = 0 to y = Y gives
u
Y
Y
Y

  u 
Y
2


u
dy

(

u
)


g

x  T  T dy
o
x o
  y  o
o
(6)
Y
Y
 T 

Y

(7)
u T  T dy   T  T  o   

x o
 y  o
Some of these terms can be evaluated directly from the boundary conditions. Both velocity
components are zero at y = 0, and at y = Y, if Y is sufficiently large. The fact that u, v, and T all
go to constant values for sufficiently large values of y implies that their derivatives with repspect
to y all go to zero for sufficiently large values of y. Therefore, terms in equations (6) and (7) can
be evaluated as follows.
Y
  u 
  u 
     
(u ) o  0,
,
  y  o
  y  y 0
Y
 T  T  o
Y
Y
 T 
 T 
    
 0,  
 y  o
 y  y 0
With these substitutions, equations (6) and (7) become
Y
Y

  u 
2


u
dy



g

x  T  T dy
x o
  y  y 0
o
(8)
Y
 T 


u T  T dy   

x o
 y  y 0
(9)
Now, we will assume functional forms for u and T that look qualitatively like those shown in
figure (1). Following the work of many others, we will assume the functional forms
2
u
y
y
 1   for 0 ≤ y ≤ δ,
U  
u = 0 for y >δ
(10)
T  T  y 
 1   for 0 ≤ y ≤ δ, T =T for y > δ
(11)
Tw  T   
where U is an unknown characteristic velocity and  is the unknown boundary layer thickness.
The boundary layer thickness for the momentum and thermal boundary layers is assumed to be
the same. Substituting these into equations (8) and (9) gives
2
2 2



d  2  y 
y  
U
y


U
 g x  Tw  T  1   dy
(12)
1   dy   
dx  0      



o


4

   y 
y    2 Tw  T 
(13)
Tw  T  U   1   dy  
x  0       

The integrals appearing in these equations evaluate to
2 2
  y

  1  y   dy   
       
 105

0

2

2
y


o 1    dy  3
4
  y 
 
  1  y  dy   
 0        30
 
 
Substituting these results into equations (12) and (13) gives
1 d
U


(14)
U 2   
 g x  Tw  T 
105 dx

3


Tw  T    U   2 Tw  T
(15)
x  30 

Expanding the derivatives and denoting differentiation by a prime, these become
U
(16)
2UU 'U 2 '  105
 35 g x  (Tw  T )

 2U '  U '  60
(17)
One method of solving these is to postulate that U and  will have the form of power-laws in x.
That is, we assume that
 ( x )  ax b , and U ( x )  cx d
These give the appropriate conditions at U(0) = (0) = 0. So if we can find values of the four
constants a, b, c, and d, then these will be the solutions to the problem. Substituting these into
(16) and (17) gives
35(3cx ( d b )  g x a 2  (Tw  T ) x b )
2
( b  2 d 1)
(18)
ac (2d  b) x

a
(19)
a 2 c(b  d ) x ( 2bd 1)  120
Equation (19) can only be satisfied for all values of x if 2b + d – 1 = 0, so we have d = 1 - 2b.
Substituting this into (18) gives
35(3cx (13b)  g x a 2  (Tw  T ) x b )
(20)
ac 2 (2  3b) x (13b ) 
a
Equation (20) can only be satisfied for all x if 1-3b = b, so we have b = ¼ and d = ½.
Substituting these into equations (18) and (19) gives equations that are independent of x.
a 2c 2  28(3c  g x a 2  (Tw  T ))
(21)
2
(22)
a c  80
These equations can be solved for a and c to give the rather complicated expressions
 5 20   21  

a  2
 7 g x  (Tw  T ) 
1/ 4
(23)
 35 g x Tw  T  

c  4
20


21



So, the solutions for (x) and U(x) are
(24)
 5 20   21 x 

 ( x)  2
 7 g x  (Tw  T ) 
(25)
1/ 4
1/ 4
 35 g x Tw  T x 

U ( x)  4
 20   21 
These can be re-written in terms of the Prandtl number and the Grashof number as
1/ 2
 520  21 Pr  

 ( x)  2 x
2
7
Pr
Gr
x


1/ 4
 0.952  Pr 
 3.94 x

Pr 2


(26)
1/ 4
Grx1 / 4
1/ 2
(27)
4 Pr  35Grx 
  Grx Pr 2 

(28)
U ( x) 

  5.164 
x  20  21 Pr 
x  0.952  Pr 
What we are generally most interested in is the overall heat transfer from the plate to the fluid. A
heat transfer coefficient (for a particular vertical position x) can be defined by
q w
(29)
hx 
Tw  T
where q w is the heat flux from the wall to the gas, which is
1/ 2
 1/ 4
2k (Tw  T ) 2k (Tw  T ) 
 dT 
Pr 2

Grx

(30)
q w  k 



3.94 x  0.952  Pr 
 dy  y 0
A local Nusselt number (dimensionless heat transfer coefficient) is then defined as

 1/ 4
xh
xq w
Pr 2
Grx
(31)
Nu x  x 
 0.508
k
k (Tw  T )
 0.952  Pr 
This is the approximate theoretical expression for the Nusselt number that is presented in the
main part of the notes. For Pr=0.72 this prediction is within 6% of the result from the exact
similarity solution derived by the differential analysis of the equations presented below.
Appendix 3: Differential Method of Analyzing the Free Convection Boundary
Layer Equations
The governing equations for the free convection boundary layer on a vertical plate are:
u  v

0
(1)
x y
u
u   2 u
(2)
u


 g x  (T  T )
x
y  y 2
T
T
2T
(3)
u
v
 2
x
y
y
with boundary conditions:
T = Tw at y = 0
T = T for y  
u = v = 0 at y = 0
u = v = 0 for y  
Equation 1 will automatically be satisfied if we define a stream function  such that


u
, and v  
y
x
then
u v  2  2



0
x y xy yx
It is convenient to also define a dimensionless temperature as
T  T

Tw  T
With these definitions, equations (2) and (3) become
  2   2   3


 g x  (Tw  T )
(4)
y xy x y 2
 y 3
   
 2
(5)

 2
y x x y
y
Now we will look for a similarity solution to these equations. We will define a stretching
function (that only depends on x) and replace the y coordinate by a similarity variable. That is,
we define
 = yH(x)
where H(x) is the stretching function and  is the new independent variable (the similarity
variable). Since H is independent of y, when we apply the chain rule for differentiation with
respect to y, we obtain
  


H
y y 

Making this substitution in equations (4) and (5) gives
3
  2
  2
3   
H2
H2

H
 g x  (Tw  T )
(6)
 x
x  2
  3
 
 
 2
(7)
H
 H 2 2
 x
x 

Now, we can apply the separation of variables method to convert these partial differential
equations into ordinary differential equations. We will assume that we can write the stream
function as the product of two functions that each only depend on one variable.

(8)
 ( x, )  F ( )G ( x)

Substituting (8) into equations (6) and (7) gives
H
2
2
2
2
3
   2  dF  dG    2
dG d 2 F
3   d F
  H G



(9)
   H FG

H
G
   d 3  g x  (Tw  T )
dx d 2

 d  dx   
 

dF  
dG 
 2
(10)
HG
 HF
 H 2 2

d x 
dx 

Our definition of  = yH(x) was made so that we could assume that there is a similarity solution.
That is, we assume that the scale factor H(x) contains all of the x dependence of the temperature
field. Therefore, the partial derivative of  with respect to x at constant  is zero, and we can
drop the first term of equation (10). Doing this, and dividing (9) by gx(Tw - T), as well as
dividing (10) by H2 gives
2
  dF  2 dG

H 2G
dG d 2 F
d 3 F 

 


(11)
F
H
  0
dx d 2
d 3 
   g x  (Tw  T )   d  dx
d 2
F dG d
(12)

0
2
d
H dx d
dG
We can eliminate the x dependence in (12) by setting the stretching factor H equal to
.
dx
dG
H
dx
Substituting this into both (11) and (12) gives
2
2
3

G
d 2 F d 3 F 
 dG    dF 
 
  F
(13)

  0

 
d 2 d 3 
   g x  (Tw  T )  dx    d 
 2 F 
(14)

0
 2  
From equation (13) we can now write
2
2
3

G
d 2 F d 3 F 
 dG    dF 
  F
(15)
   


 
d 2 d 3 
   g x  (Tw  T )  dx    d 
Since  is written here as the product of a function that only depends on x and a function that
only depends on , and we have assumed that  is independent of , the function of x must be
constant. For convenience, we can set it equal to 1. So, we write
2

G
 dG 
 

 1
   g x  (Tw  T )  dx 
3
(16)
This is a separable ODE for G. Taking the cube root of it and rearranging gives
  2

G
 

    g x  (Tw  T ) 


1/ 3
dG  dx
(17)
Integrating this once with the condition G = 0 at x = 0 gives
2

3    
G4

 
4     g x  (Tw  T ) 


Solving this for G gives
1/ 3
x
(18)
1/ 4




  43 x 3 g x  Tw  T  
G
(19)

2





 





This is usually written in terms of the dimensionless Grashof number (Gr), instead of x, where
x 3 g x  (Tw  T )
(20)
Grx 
2



 
In terms of the Grashof number, G and H are then
4
G 
3
3/ 4
Gr 1 / 4
1/ 4
(21)
dG  3  Gr 1 / 4
 
(22)
dx  4 
x
Now all that remains is to solve the ordinary differential equations, (13) and (14), for the
functions F and . Introducing the dimensionless Prandtl number (Pr = /()), and denoting
differentiation with respect to  (the only remaining independent variable) by a prime, these
equations are
(23)
( F ' ) 2  FF ' ' F ' ' '  0
(24)
   Pr F   0
The boundary conditions for these equations come from the original boundary conditions on u, v,
and T. These are as follows:
T = Tw at y = 0, which implies that
 = 1 at  = 0
(25)
T = T for y  , which implies that
 = 0 for   
(26)
u = 0 at y = 0, which implies that
F = 0 at  = 0
(27)
v = 0 at y = 0, which implies that
F = 0 at  = 0
(28)
u = 0 for y  , which implies that
F = 0 for   
(29)
Assuming, for the moment, that we could solve these equations to obtain F() = f1() and () =
f2(), we would want to convert these to the quantities of interest in the physical coordinates, x
and y. For , we simply transform back to the x, y coordinates:
and
H
  3 1 / 4 1

1/ 4

 ( x, y )  f 2 ( Hy )  f 2   
Gr  y 
 4  x



We reconstruct the stream function as
 ( x, y ) 
(30)

F ( Hy )G( x)

(31)
Using this, the velocities are then
  3 1 / 4 1

1/ 4

(32)
f1 '   
Gr  y 
 4  x



1/ 2
1/ 4
 3  1


 dG 1   3

1/ 4

(33)
v
 HF
 2  Grx  f1   
Gr  y 
 4  x

x
 dx x   4



What we are generally most interested in is the overall heat transfer from the plate to the fluid. A
heat transfer coefficient (for a particular vertical position x) can be defined by
q w
(34)
hx 
Tw  T
where q w is the heat flux from the wall to the gas, which is
u


 dF 1   4

H
 HG

 Grx 
y

 d x   3

1/ 2
 dT 
 d 
 d 



q w  k 
 k (Tw  T )
 k (Tw  T ) H 
 dy  y 0
 dy  y 0
 d   0
So, the heat transfer coefficient is
 d 

hx  kH 
 d   0
A local Nusselt number (dimensionless heat transfer coefficient) is then defined as
1/ 4
xhx
 d 
 3   d 


Nu x 
  xH 
  Gr  
k
 4   d   0
 d   0
(35)
(36)
(37)
But, alas, all of this does us no good, since we cannot analytically solve the nonlinear coupled
ODE’s for F() and (). However, for a particular value of the Prandtl number, we can solve
them numerically. Note that these equations only depend on the Prandtl number (the ratio of
kinematic viscosity to the thermal diffusivity). The simplest way to solve the ODE’s is probably
to use a shooting method. If we had only boundary conditions at one side ( = 0), then the
ODE’s would form an initial value problem. Matlab, Maple, and other packages can readily
solve such a problem. It would be more difficult to solve the boundary value problem that we
actually have. So, what we will do is replace the boundary conditions at  with additional
boundary conditions at  = 0. We will then solve the problem for different conditions at  = 0
until we find conditions that cause the actual boundary conditions at  to be satisfied. In this
case, instead of specifying that  = 0 at , we will specify a value for  at  = 0. Likewise,
instead of specifying that F = 0 at , we will specify a value for F at  = 0.
To make it easier to implement the solution of these equations using numerical analysis software,
we can convert the system of one second order ODE and one third order ODE into a system of 5
first-order ODE’s. To do so, we define W = F, X = F, and Y = . These definitions provide
three additional equations (but we have introduced three additional unknowns as well). The
system of five 1st-order ODE’s can then be written as:
(38)
F  W
(39)
W  X
2
(40)
X   W  FX  
(41)
  Y
(42)
Y    Pr FY
We have three known initial conditions (F(0) = 0, W(0) = 0, and (0)= 1). To solve these using
the shooting method, we will guess values for X(0) and Y(0) and vary these guesses until we
satisfy the actual boundary conditions ( = 0 for   , and W = 0 for   ). This variation
can also be done automatically using a minimization program to minimize the absolute values of
 and W for large values of . From equation (37), we have
1/ 4
1/ 4
 3   d 
3

Nu x Grx1 / 4    
   W (0)
(43)
 4   d   0
4
The right-hand-side of this expression only depends on the Prandtl number. The numerical
solution (using the shooting method as described above) gave the following values for this
expression.
Pr
1
Nu x Grx
4
0.01
0.0546
0.1
0.157
0.72
0.348
1.0
0.392
10
0.818
100
1.52
1000
2.74
These results are all around 2% lower than those cited by Kays and Crawford (reference 9). They
used a slightly different similarity transform but, in principle, the results should be the same. A
plot of the five functions (F, W, X, , and Y) is shown below for Pr = 0.7, which is roughly the
value for air at room temperature.
Appendix 4: LabView program for recording temperatures
The front panel of the LabView virtual instrument (data acquisition program) for this experiment
is shown below:
System Settings (inputs)
System Readings (outputs)
The controls listed under system settings are items that you enter by typing numbers in the boxes
or by clicking on the button. You enter a time to wait between temperature readings (the
minimum is around 5 milliseconds) and a number of temperature readings to average before
displaying the result. The time between data points that are displayed is the product of the
waiting time and the number of points. For example, if you wait 10 seconds between readings
and average 100 readings, you will record one data point about every 1000 milliseconds, or 1
data point per second. There is also a control to enter the position of the movable thermocouple
(TC 8). This information is not used by the data acquisition program, but is written to the output
file along with the measured temperatures. The three indicators under system readings show the
temperatures measured by the movable thermocouple (TC 8), the fixed thermocouple that is
measures the ambient temperature (TC 7), and whichever other thermocouple has been selected
using the multi-position switch. When you start the program, it will prompt you to enter the
name of a file in which the data should be written. When you enter a name, the program will
create and open that file. After that, each time you click on the button under system settings, the
movable thermocouple position and the three temperature measurements will be written to the
output file. Note that the file will not include any information on which thermocouple was
selected to be read by channel 3, so you will need to make note of that separately. So, to do the
experiment, you first start the vi and enter a name for the output file. Then, starting with the
movable thermocouple just touching the wall, you enter the position of the thermocouple, click to
write the measurements to the file, and then move the thermocouple and repeat until the
temperature measured by the movable thermocouple approaches the temperature measured by the
ambient thermocouple.
Now that we have seen how to use the virtual instrument, we will try to understand the diagram
that was used to build this virtual instrument. It is shown on the following page. We see that
this program contains three levels of loops. The outer 'while loop' is an infinite loop. That is, it
will keep looping forever, or until the stop button on the front panel is pushed. The conditional
for continuing the loop is connected to the boolean constant 'false' through the boolean operator
'not'. This is equivalent to saying (do while (not.false) ...) The 'not false' always produces the
result 'true', so the condition for repeating the loop is always satisfied. Infinite loops are
generally bad programming practice, and this structure might be one of the first things to
eliminate if one was to try to improve this program.
Within the infinite while loop is another while loop. Wired to its conditional for continuing the
loop is the boolean control from the front screen that is used to write measurements to the output
file. The control is set so that its default state is 'true', and so that clicking on it sets it to 'false'
only temporarily. It remains 'false' the first time it is read, and after that it reverts to 'true'. As
long as this control is set to 'true', the inner while loop will repeat. When we click on it, it is
temporarily set to 'false'. This causes the program to leave the inner while loop and go to the
outer loop, where a set of data is written to the output file. The control then reverts to 'true', and
the inner loop repeats until the button is pressed again.
Within the inner 'while' loop is a 'for loop'. It is within this loop that the data acquisition actually
takes place. The number of times to repeat this loop before exiting is given by the digital control
that is labeled 'Number of Readings to Average'. It is wired to the input that specifies the number
of times to go through the loop. Each time through the loop, the program reads the three
temperatures from the board, and then waits a specified number of milliseconds. To read a
temperature, the program uses the 'AI ONE PT' vi to get a voltage reading from the data
acquisition board to which the thermocouples are connected. The inputs to this vi, from top to
bottom, are the device number to read from, the channel of that device to read from, the
maximum value expected and the minimum value expected. Thus, the top 'AI ONE PT' shown
above reads channel 1 of device 1, and expects a reading of between -0.0001 and 0.0001 volts.
The thermocouple that is placed in the hot water bath is physically connected to the input
terminals of this channel. Likewise, the next 'AI ONE PT' shown reads channel 0 of device 1,
and expects a reading between 0.00 and 0.50 volts. Channel zero is connected to an integrated
circuit temperature sensor on the data acquisition board. This temperature sensor provides the
known temperature that is required for us to use 'cold junction compensation' to obtain a
temperature from the thermocouple voltage. This is discussed in more detail in the lecture on
'your friend the thermocouple'. The voltage reading from each of the three thermocouples is sent
to a vi called 'THERMOLINEAR'. This is a vi, provided with LabView, that converts a
thermocouple voltage to a temperature. As inputs it requires the thermocouple voltage, a voltage
from the temperature sensor that is being used for cold junction compensation, a specification of
the thermocouple type, and a specification of the type of temperature sensor being used for cold
junction compensation. The voltage inputs are wired to the appropriate outputs of the 'AI ONE
PT' vi's. The other inputs are specified as constants - we are using type T thermocouples for the
temperatures we want, and an IC Sensor for the reference junction temperature. The outputs
from the 'THERMOLINEAR' vi's are the measured temperatures. These are sent out of the for
loop. You will notice that when the wires leave the loop, they become thicker. This is because
the loop has 'auto-indexing' enabled. This means that it takes the value at each iteration and adds
it to a list (a one-dimensional matrix, or vector), so that when the loop finishes, the output is a list
of numbers whose size corresponds to the number of iterations that were completed. Each list of
measured temperatures goes to a 'MEAN' vi. This is another vi that is provided with LabView.
This vi simply takes the average of the list of numbers. Its output is therefore a single number, as
indicated by the narrow line leaving it. The outputs from the 'MEAN' vi's are sent to the digital
indicators that are shown on the front panel. They are also sent out of the 'while' loop for storage
in the data file.
When the data leaves the inner while loop, auto-indexing is not used. The thin lines leaving the
loop indicate scalar values. These are the temperatures read the last time the while loop was
executed. The value entered for the thermocouple position also leaves the while loop. These
values from the last time the inner while loop executes are the next set of values displayed after
the button on the front panel is clicked. The values are sent to the 'build array' vi, which puts
them into a 1-dimensional array. This array is sent to the 'write to spreadsheet' vi, which appends
the four numbers to the end of the output file. The name of the output file comes from the
'open/create/replace file' vi, which is the first vi called when the program executes.
This discussion is intended to give you some idea how the data acquisition program (or virtual
instrument) works. To understand it in detail, you will have to work through the diagram itself.
All of the vi's have online help that explains what their inputs and outputs are, what they do, and
how they do it.