Chapter 8 Internal Forced Convection Flashcards
What is the advantage of circular pipes over non circular pipes
they can withstand large pressure differences between the inside and the outside without undergoing any significant distortion
for fixed surface area gives the most heat transfer for the least pressure drop
What it the fluid velocity maximum in a pipe and why
at the centre as the no slip condition makes it zero at the wall increases inwards
As dP/dx is constant what does this mean for curvature
v*d^2U/dy^2 = dP/dx = const therefore curvature is constant, making it a parabola
What is the average velocity and why is it used
average velocity remains constant in incompressible flow when the cross section area is constant
used for convenience
For fully developed laminar flow in a pipe Vavg is defined as the
the average speed through a cross section, equal to half the maximum velocity
Equation for finding V avg =
2/R^2 * integral 0 to R of the velocity profile *r *dr
derive the equation for V avg
use equations for mass flow rate rho V avg * Ac = integral of the cross sectional area rhovelocity profiledAc see book
How is the average temperature calculated
found by averaging the energy within a flow profile
Derive the equation for Tm (or Tavg)
Start with energy flow in pipe E.fluid = m.cpTm = integral from m. of cpT(r) * delta m.
Equation for Tm =
2/Vavg*R^2 * integral from 0 to R of T(r) * u(r) * r * dr
Transition does no occur…
suddenly occurs over some range of velocity where the flow fluctuates between laminar and turbulent flows before it becomes fully turbulent
What does the transition region depend on
reynolds number as well as the degree of disturbance of the flow by surface roughness, pipe vibrations and the fluctuations in the flow
What does the hydraulic diameter =
Dh = 4*Ac/p
What does the hydraulic diameter = in cricular pipes
D
In a channel what does the the hydraulic diameter =
Dh = 4*ab/2a + b, perimeter is only bit touching the fluid
only the wetted perimeter is included
When is a flow fully turbulent
Re > 10,000
When is the Dh used
when calculating Re, Nu and Pr
What is the entrance region
area where the velocity profile develops
Draw the velocity entrance region
see book
What is the boundary layer region,
the viscous effects and the velocity changes are significant
What is the irrotional core flow region
the region where the frictional effects are negligible and the velocity remains essentially constant in the radial direction
fully inviscid and irrotional
what is the hydrodynamic entrance region
the region from the pipe inlet to the point at which the velocity profile is fully developed
What is the hydodynamic entrance length Lh
the length of the hydrodynamic entrance region
What is the hydrodynamic fully developed region
the region beyond the entrance region in which the velocity profile is fully developed and remains unchanged
What is the velocity gradient strongest
just after the fluid enters the pipe
what region does the velocity profile no longer depend on x
fully developed region
where can bernouillis be used
in the irrotational core region and not experiencing viscous effects
At what temperature are fluid properties evaluated at
the bulk mean fluid temperature, which is the arithmetic average of the mean temperatures at the inlet and exit Tb= Tmi + Tme /2
What is the thermal entrance region
the region of flow over which the thermal boundary layer develops and reaches the tube centre
What is the thermal entry length
the length of the thermal entrance region
Thermally developing flow
flow in the thermal entrance region, where the temperature profile develops
Fully developed flow
the region in which the flow is both hydrodynamically and thermally developed
thermally fully developed region
the region beyond which the thermal entrance region in which the dimensionless temperature profile remains unchanged
In the core region of the thermal entrance region what = 0
dT/dy ie there is no heat transfer as no temperature difference
What happens in the thermally fully developed region of a tube
the local convection coefficient is constant (does not vary with x)
what is higher in the entrance regions of a tube
the pressure drop and heat flux
what does an entrance region always increase
the average friction factor and heat transfer coefficient for the entire tube
What partial derivative is used to describe hydrodynamically fully developed flow
d u(r,x)/ dx = 0 -> u = u(r)
What partial derivative is used to describe thermally fully developed flow
d/dx *((Ts(x) - T(r,x))/(Ts(x)-Tm(x)) = 0
Derive an equation for finding h using the surface energy balance
q.s = hx(Ts -Tm) = k * pardT/pardr at r=R
gives hx = (k*pardT/pardr @ r=R) / Ts - Tm
Prandtl number =
kinematic viscousity / thermal conductivity
What does a high Pr number mean for the entrance region
high kinematic viscosity therefore momentum diffuses fast therefore velocity entrance length is small
What does a low Pr number mean for the entrance region
high thermal conductivity therefore thermal entrance length is small
Pr ratios =
v/alpha = partial v^2/ partial T^2 = L temp/ L Vel
what does the thermal profile look like once fully developed
parabola, the shape will remain constant, this is why for the equation
d/dx *((Ts(x) - T(r,x))/(Ts(x)-Tm(x)) = 0
we take away heat to keep it constant
What dimensionless number is much higher in the entrance region
Nu and thus h values are much higher in the entrance region
When does the Nu number reach a constant
at a distance of less than 10 diameters meaning we can assume flow fully developed after x>10D
Lv or Lh laminar approx =
0.05 Re D
Lt laminar approx =
0.05 Re Pr D = Pr Lh laminar
Lv turbulent approx =
Lt turbulent approx = 10D
How does the local nusselt number vary between constant surface temperature and constant heat flux
it takes longer for constant heat flux to return to the local Nu number to a constant level
How can thermal conditions at the surface by approximate
constant surface temperature Ts = const
Constant surface heat flux qs =const
Rate of heat transfer =
Q. = m. cp (Te - Ti)
Surface heat flux =
q.s = hx (Ts - Tm) where hx is the local heat transfer coefficient
Heat transfer to a fluid flowing in a tube is
= to the increase in energy of the fluid
when does constant surface temperature occur
when a phase change process such a boiling or condensation occurs at the outer surface of a tube
when does constant surface heat flux condition occur
when the tube is subjected to radiation or electric resistance heating uniformly from all directions
Can we have Ts = const and qs = const
No
By considering constant surface heat flux derive mean fluid temperature at tube exit
Q. = q.s As = m.cp (Te - Ti) Te = Ti + q.s As/m. cp
By considering constant surface heat flux derive mean surface temperature
q.s = h (Ts - Tm) -> Ts = Tm + q.s/h
What does Ts = Tm + q.s/h mean for Ts
Ts grows linearly as q is constant
Draw a temperature vs x graph for Ts and Tm
See book
Prove that the shape of the temperature profile remains unchanged in the fully developed region of a tube subject to constant surface heat flux
see book
With constant surface temperature what can not constantly increase
the temperature of the fluid, it will max out at Ts
Equation for rate of heat transfer to or from a fluid flowing in a tube with constant Ts
Q. = hAs deltaTavg = h As (Ts - Tm)avg
What does delta Tavg approx =
for constant Ts
Ts - Tb
from approx = change in Ti + Change in Te /2 = (Ts - Ti) + (Ts - Te) /2 = Ts - (Ti + Te)/2 = Ts - Tb
Whats the issue with using arithmetic mean temperature difference
Ts
we assume that the mean fluid temperature varies linearly along the tube which is hardly ever the case when Ts = const
Te for constant Ts =
Ts - (Ts - Ti)exp(-hAs/m.cp)
Derive the equation for Te
see book
What does NTU stand for
number of transfer units, a measure of the effectiveness of the heat transfer system
says how fast we get temp of fluid reaching temp of surface
for NTU = 5
Te = Ts, limit of heat transfer is reached
What does a small value of NTU
more opportunities for heat transfer
what is delta Tln
An exact representation of the average tempearture difference between the fluid and the surface
What is the error in using the arithmetic mean temperature when difference in change in Te and change in Ti is 40 percent
less than 1%
what does an NTU of greater than 5 represent
that fluid flowing in a tube will reach the surface temperature at the exit regardless of the inlet temperature
Heat transfer for Ts Q. =
Q. = hAs * delta Tln
What does delta Tlin =
(Change in Te - Change in Ti)/ ln(Change in Te)/(Change in Ti)
what does NTU =
hAs/m.cp
what does NTU =
hAs/m.cp
Constant surface temp what does Nu =
3.66 = hD/k
For turbulent flow in tubes what is the colburn equation Nu =
0.023 Re^0.8 Pr ^1/3 0.7 <= Pr <= 160 and Re>10000
For turbulent flow in tubes what is the chilton colburn analogy Nu =
0.125 f * Re*Pr^1/3
where f = 0.184 Re^-0.2
For turbulent flow in tubes what is the dittus boelter equation Nu =
0.023 Re ^0.8 Pr ^n where n = 0.4 for heating and 0.3 for cooling
When variation in properties is large due to a large tempearture difference Nu
0.027 Re ^0.8 Pr ^1/3 (mu/mu(s))^0.14 0.7 <= Pr <= 17600 and Re>10000
all properties evaluated at Tb expect mu(s) which is evaluated at Ts
