AME 331, Prof. P. Ronney

Midterm Exam #2 Study Guide

April 17, 2008

The
second midterm exam will be similar to the first midterm - unlimited open
book. You may use any reference
materials you want.

The exam may cover any material
through the end of chapter 7 (natural convection) but will emphasize material
covered since the first midterm, i.e. convection (chapters 5 – 7,
homework sets 4 – 6). This
includes:

¥ Basic fluid mechanical concepts

¥ Viscosity

¥ Reynolds and Prandtl number

¥ Transition to turbulence

¥ Navier-Stokes equation (F = d(mv)/dt for a fluid)

¥ Continuity (mass conservation) equation

¥ Energy equation with convection term

¥ Forced convection – external flows

¥ Flow over a flat plate

¥ Viscous boundary layer

¥ Thermal boundary layer

¥ Laminar flow

¥ Boundary layer thickness

¥ h and Nu

¥ Constant surface temperature and constant surface heat flux

¥ Local vs. averaged h

¥ Film temperature

¥ Turbulent flow over a flat plate

¥ Boundary layer thickness

¥ h and Nu

¥ Constant surface temperature and constant surface heat flux

¥ Cross-flow over a cylinder

¥ Sphere

¥ Forced convection – internal flows

¥ Flow inside pipes

¥ Entrance vs. fully developed flow

¥ Friction factor – smooth vs. rough tubes

¥ Moody diagram

¥ Bulk temperature (T_{b})

¥ Conservation of energy along the tube length (constant T_{w}):
(T_{w} – T_{b}(L))/(T_{w} – T_{b}(0))
= exp[(-h¹D/C_{P})L]

¥ Hydraulic diameter = 4A/P – use as diameter (D) when
no specific relation for the channel shape of interest is available

¥ Overall heat transfer coefficient (U) – combined
effect of convective thermal resistance from the inside fluid to the inside
pipe wall, conductive resistances across the wall, convective resistance from
the outside pipe wall to ambient fluid outside pipe

¥ Buoyant (aka natural, free) convection

¥ Grashof and Rayleigh numbers

¥ Vertical flat plate

¥
Constant temperature

¥
Boundary layer thickness

¥ d/x = C(Gr_{x}/4)^{-1/4}

¥ d_{max}, i.e.
location where u is maximum: C =
-0.25(log_{10}Pr)^{ }+ 0.9579

¥ d_{99}, i.e.
location where u = 0.01u_{max}: C = -0.7735(log_{10}Pr)^{3}
+ 3.8176(log_{10}Pr)^{2} – 1.1712(log_{10}Pr) +
5.5023

¥
Constant wall heat flux: use Gr_{L}^{*} ¼ gb(q/A)L^{4}/kn^{2}

¥ Horizontal plates

¥ Horizontal cylinders

¥ Spheres

¥ Enclosures

¥ Summary of convection

¥ Energy equation has extra term (compared to conduction only)
involving fluid velocity

¥ Need Navier-Stokes equations (x-momentum conservation,
y-momentum conservation) plus continuity (mass conservation) for 3 unknowns (x-velocity,
y-velocity, pressure) – leads to complicated phenomena (shocks,
turbulence, etc.)

¥ Energy conservation adds one more equation and one more
unknown (T)

¥ To calculate heat transfer coefficient (h), all we really
need to know is ¶T/¶y at surface, but in order to get that we have to solve NS
+ continuity + energy equations!

¥ At sufficiently high Re or Gr, the effect of viscosity is
confined to a thin layer near the surface (boundary layer) which ÒsimplifiesÓ
the analysis

¥ Still, we usually we resort to empirical relations typically
of the form Nu = C Re^{n}Pr^{m} (forced) or Nu = C Ra^{n}Pr^{m}
(buoyant)

¥ At least two possibilities to consider

¥ Surface temperature (T_{w}) known, calculate surface
heat flux (q/A)

¥ Surface heat flux (q/A) known, calculate surface temperature
(T_{w})

¥ Forced convection:
free-stream velocity (u_{°}) known, calculate entire flow field
u(x,y), v(x,y) (u = x-component of velocity, v = y-component of velocity), plug
into energy equation

¥ Buoyant convection: important in most large fluid systems
when there is little or no forced flow - complicated interaction between fluid
mechanics and heat transfer because momentum and energy equations coupled

¥ Heat transfer causes DT

¥ DT causes buoyant
force

¥ Buoyant force causes fluid flow

¥ Fluid flow causes heat transfer

¥ Etc., etc.

¥ Grashof or Rayleigh number describes importance of buoyant
flow; generally Gr or Ra ~ Re^{2}, where Re is NOT based on u_{°}
but rather the velocity induced by buoyancy; frequently (but not always)
properties that like Re^{n} in forced flow will scale as Gr^{n/2}
or Ra^{n/2} in buoyant flow

¥ Be careful – before plugging numbers into a formula,
consider

¥ Forced or buoyant?
May need to check both and see which is more important

¥ Geometry (flat plate, cylinder, enclosure, etc.; if buoyant,
horizontal or vertical)

¥ Laminar or turbulent (depends on Re in forced convection; Gr
or Ra in buoyant convection)

¥ Constant wall temperature or constant heat flux

¥ Applicable range of Gr or Ra

¥ Film ( )_{f}, surface ( )_{w}, bulk ( )_{b}
or free-stream ( )_{°} temperature – which to use for property
evaluation?

¥ Sometimes problems also include a solid-phase conductivity
(e.g. problems involving heat transfer across a pipe wall) but NEVER use the
solid-phase conductivity to compute convective heat transfer between the fluid
and the solid surface; use solid-phase conductivity to calculate heat transfer **across** the wall

Note: since empirical relations are important
in convection problems, it may be useful to prepare a table with all relevant
formulas and their restrictions, or at least Xerox the pages out of the text
with the formula summaries. Also
be sure you know the definition of Nusselt, Reynolds, Grashof, Rayleigh and
Prandtl numbers.

__Last yearÕs second midterm (average 68, high
88)__

__ __

__(Note
this was a 55 minute exam because the class met 3 times per week)__

__ __

__Problem
#1 (40 points total)__

__ __

A
smooth pipe 1 m long has a constant surface temperature of 90ûC. Water flows into the pipe at 27ûC, that
is, T_{b}(0) = 27ûC. Water
properties at 27ûC: r = 997 kg/m^{3};
C_{p} = 4179 J/kgK; k = 0.613 W/mK; µ = 8.55 x 10^{-4} kg/m_{
}s; n = 8.58 x 10^{-7}
m^{2}/s; Pr = 5.83.

(a)
(10 points) If the water mass flow rate is 0.01
kg/s and the pipe diameter is 0.03 m, what is the bulk temperature (T_{b}(L))
at the exit of the pipe?

(b) (10 points) If the water mass flow rate is 0.01 kg/s and the pipe
diameter is 0.003 m, what is the bulk temperature (T_{b}(L)) at the
exit of the pipe?

(c) (10 points) If the water mass flow rate is 0.001 kg/s and the pipe
diameter is 0.03 m, what is the bulk temperature (T_{b}(L)) at the exit
of the pipe?

(d) (10 points) If instead the pipe is 1000 m long, for set of conditions,
(a), (b) or (c), will the bulk temperature (T_{b}(L)) at the exit of
the pipe be highest? (You donÕt
have to re-do the calculations in parts (a) – (c), just state your answer
and the reason for it.)

With
this L, is very large, thus exp(-) Å 0, thus T_{b}(L) Å 90ûC for all 3 cases.

__ __

__Problem
#2 (40 points total)__

A large covered square
pan of warm chili (enough to feed the whole class) is 20 cm tall and has a 40
cm x 40 cm cross-section. The
vertical sides and top cover lose heat to the surrounding air at 27ûC by
buoyant convection only. The
bottom rests on a 60 cm circular disk, which in turn sits on table made of wood
(k = 0.25 W/mK) at 27ûC. The pan,
cover and disk are made of very conductive steel, so all these surfaces have
the same temperature T_{s} = 50ûC. Air properties at 27ûC: r = 1.161 kg/m^{3}; C_{p} = 1007
J/kgK; k = 0.0263 W/mK; µ = 1.85 x 10^{-5} kg/máás; n = 1.59 x 10^{-5} m^{2}/s; Pr =
0.707.

(a)
(15 points) What is the heat transfer coefficient
from the vertical sides of the pan to the air?

(b)
(15 points) What is the heat transfer coefficient from the top cover of
the pan to the air?

(c)
(10 points) What is the total heat loss (in Watts) from the pan,
including the top cover, sides and bottom?

__ __

Bottom:
square with side D to semi-infinite solid; q = kS(DT); S = 2D (Not in your text; I used a different
textboox last time.)

q_{total} = q_{top} + q_{sides}
+ q_{bottom} = h_{top}A_{top}DT + h_{sides}A_{sides}DT + k_{table}(2D)(DT)

= (4.38 W/m^{2}K)(0.4 m)(0.4 m)(23K)

+ 4(4.97 W/m^{2}K)(0.2 m)(0.4 m)(23K)

+ (0.25 W/mK)(2)(0.6 m)(23K)

= 59.6 W

__ __

__Problem
#3 __ (20 points)

Explain
the difference between convection heat transfer analyses for *constant
surface temperature* (T_{w})
vs. *constant surface heat flux*
(q/A). In particular, answer the
following questions: (a) Are the governing equations for mass,
momentum and energy conservation the same? (b) Are the
boundary conditions the same?
(c) Are the resulting
correlations of Nusselt number to Prandtl and Reynolds (or Grashof) number the
same? (d) What is different about the use of
relations for constant surface heat flux for buoyant convection than forced
convection?

For convection problems, the heat transfer
coefficient (h) is defined by the relation q/A = h(DT).
Thus, we can consider two cases:
one where DT = T_{s}
– T_{°} is known and q/A is computed once we determine h; the
other case where q/A is known and DT = T_{s} – T_{°} is computed once we determine
h. Concerning specific questions
asked:

(a)
Governing equations
are exactly the same – mass, momentum and energy still need to be
conserved!

(b)
Boundary conditions
are different:

á
Constant surface
temperature: T = T_{s} =
constant at y = y_{surface}

á
Constant heat
flux: q/A = k_{fluid}(¶T/¶y)
= constant at y = y_{surface}

(c)
No, the resulting
correlations Nu = Nu(Re, Pr) or Nu = Nu(Gr, Pr) are different; in the case of
forced flow, however, they are ^{frequently} very similar. For example, for laminar forced flow
over a flat plate, Nu_{x} = 0.332 Re_{x}^{1/2}Pr^{1/3}
for constant T_{s} and Nu_{x} = 0.453 Re_{x}^{1/2}Pr^{1/3}
for constant q/A (in this particular case the two correlations are different
only by a constant factor of about 1.36; for turbulent flow the difference
between constant T_{s} and constant q/A is even smaller.)

(d)
In the case of
buoyant convection with q/A = constant we canÕt use Gr ¼ gb(DT)x^{3}/n^{2} as the driving force for heat
transfer since DT is unknown (DT is what weÕre trying to determine). instead we have to use Gr^{*} ¼ gb(q/A)x^{4}/kn^{2} so that we have a quantity using only known
parameters. (This was not an issue
for forced convection, because in that case the driving force is Re, which
doesnÕt depend on DT, thus Re can be
used in Nu(Re,Pr) correlations for both constant T_{s} and constant q/A
cases.)