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Fundamentals of Heat and Mass Transfer,
Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera,
David
P. DeWitt,
John Wiley & Sons, Inc.
•Chapter 1: Introduction
Conduction Heat Transfer
•Chapter 2: Introduction to Conduction
•Chapter 3: 1D, Steady-State Conduction
•Chapter 4: 2D, Steady-State Conduction
•Chapter 5: Transient Conduction
Convection Heat Transfer
•Chapter 6: Introduction to Convection
•Chapter 7: External Flow
•Chapter 8: Internal Flow
•Chapter 9: Free Convection
•Chapter 10: Boiling and Condensation
•Chapter 11: Heat Exchangers
Radiation Heat Transfer
•Chapter 12: Radiation Processes and Properties
•Chapter 13: Radiation Exchange Between Surfaces
1 Mass Transfer
•Chapter 14: Diffusion Mass Transfer
Chapter-6
(Introduction to Convection)
2
Chapter-6: Introduction to Convection (1/2)
6.1 The Convection Boundary Layers 378
6.1.1 The Velocity Boundary Layer
6.1.2 The Thermal Boundary Layer
6.1.3 The Concentration Boundary Layer
6.1.4 Significance of the Boundary Layers
6.2 Local and Average Convection Coefficients
6.2.1 Heat Transfer
6.2.2 Mass Transfer
6.2.3 The Problem of Convection
6.3 Laminar and Turbulent Flow
6.3.1 Laminar and Turbulent Velocity Boundary Layers
6.3.2 Laminar and Turbulent Thermal and Species
Concentration Boundary Layers
3
Chapter-6: Introduction to Convection (2/2)
6.4 The Boundary Layer Equations
6.4.1 Boundary Layer Equations for Laminar Flow
6.4.2 Compressible Flow
6.5 Boundary Layer Similarity: The Normalized Boundary
Layer Equations
6.5.1 Boundary Layer Similarity Parameters
6.5.2 Functional Form of the
Solution
s
6.6 Physical Interpretation of the Dimensionless Parameters
6.7 Boundary Layer Analogies
6.7.1 The Heat and Mass Transfer Analogy
6.7.2 Evaporative Cooling
6.7.3 The Reynolds Analogy
6.8 Summary
4
Boundary Layers: Physical
Features
• Velocity Boundary Layer
– A consequence of viscous effects
associated with relative motion
between a fluid and a surface.
– A region of the flow characterized by
shear stresses and velocity gradients.
– A region between the surface
and the free stream whose
thickness ( ) increases in
the flow direction.
– Why does increase in the flow direction?
– Manifested by a surface shear
stress that provides a drag
force (FD).
– How does vary in the flow
direction? Why?
Thermal Boundary Layer
– A consequence of heat transfer
between the surface and fluid.
– A region of the flow characterized
by temperature gradients and heat
fluxes.
– A region between the surface and
the free stream whose thickness
increases in the flow direction.
– Why does increase in the
flow direction?
– Manifested by a surface heat
flux qsʹʹ and a convection
heat transfer coefficient h.
– If is constant, how doand h
vary in the flow direction?
Distinction between Local and
Average Heat Transfer Coefficients
• Local Heat Flux and Coefficient:
• Average Heat Flux and Coefficient for a Uniform Surface
Temperature: q = ∫As q
ʹʹ
dAs
• For a flat plate in parallel flow:
Boundary Layer Transition
• How would you characterize conditions in the laminar region
of boundary layer
development? In the turbulent region?
• What conditions are associated with transition from laminar to
turbulent flow?
• Why is the Reynolds number an appropriate parameter for
quantifying transition
from laminar to turbulent flow?
• Transition criterion for a flat plate in parallel flow:
What may be said about transition
if ReL<Rex,c? If ReL>Rex,c?
• Effect of transition on boundary layer thickness and local
convection coefficient:
Why does transition provide a significant increase in the
boundary layer thickness?
Why does the convection coefficient decay in the laminar
region? Why does it increase
significantly with transition to turbulence, despite the increase
in the boundary layer
thickness? Why does the convection coefficient decay in the
turbulent region?
Boundary Layer Equations (1/3)
• Consider concurrent velocity and thermal boundary layer
development for steady,
two-dimensional, incompressible flow with constant fluid
properties and
negligible body forces.
• Apply conservation of mass, Newton’s 2nd Law of Motion and
conservation of energy
to a differential control volume and invoke the boundary layer
approximations.
Velocity Boundary Layer:
Thermal Boundary Layer:
Boundary Layer Equations (2/3)
Conservation of Mass:
In the context of flow through a differential control volume,
what is the physical
significance of the foregoing terms, if each is multiplied by the
mass density of
the fluid?
• Newton’s Second Law of Motion:
What is the physical significance of each term in the foregoing
equation?
Why can we express the pressure gradient as dp∞/dx instead of
∂p / ∂x ?
Boundary Layer Equations (3/3)
Conservation of Energy:
What is the physical significance of each term in the foregoing
equation?
What is the second term on the right-hand side called and under
what conditions
may it be neglected?
Boundary Layer Similarity (1/4)
• As applied to the boundary layers, the principle of similarity
is based on determining
similarity parameters that facilitate application of results
obtained for a surface
experiencing one set of conditions to geometrically similar
surfaces experiencing
different conditions. (Recall how introduction of the similarity
parameters Bi and Fo
permitted generalization of results for transient, one-
dimensional conduction).
• Dependent boundary layer variables of interest are:
• For a prescribed geometry, the corresponding independent
variables are:
Geometrical: Size (L), Location (x,y)
Hydrodynamic: Velocity (V)
Fluid Properties:
and
T = f (x , y , L, V , ρ , µ, c p , k , Ts ,T∞ )
Boundaryh=f(x,L,V,ρ,µ, c Layerp,k,Ts,T∞) Similarity (2/4)
• Key similarity parameters may be inferred by non-
dimensionalizing the momentum
and energy equations.
• Recast the boundary layer equations by introducing
dimensionless forms of
the independent and dependent variables.
• Neglecting viscous dissipation, the following normalized
forms of the x-momentum
and energy equations are obtained:
Boundary Layer Similarity (3/4)
How may the Reynolds and Prandtl numbers be interpreted
physically?
• For a prescribed geometry,
The dimensionless shear stress, or local friction coefficient, is
then
What is the functional dependence of the average friction
coefficient?
Boundary Layer Similarity (4/4)
For a prescribed geometry,
The dimensionless local convection coefficient is then
What is the functional dependence of the average Nusselt
number?
How does the Nusselt number differ from the Biot number?
Reynolds Analogy (1/2)
• Equivalence of dimensionless momentum and energy
equations
for negligible pressure gradient (dp*/dx*~0) and Pr~1:
Advection terms Diffusion
• Hence, for equivalent boundary conditions, the solutions are
of the same form:
u * = T *
∂u * = ∂T
*
∂y * ∂y* *
= 0
*
=0 y y
C f
Re
2 = Nu
Reynolds Analogy (2/2)
With Pr = 1, the Reynolds analogy, which relates important
parameters of the
velocity and thermal boundary layers, is
• Modified Reynolds (Chilton-Colburn) Analogy:
– An empirical result that extends applicability of the Reynolds
analogy:
Colburn j factor for heat transfer
– Applicable to laminar flow if dp*/dx* ~ 0.
– Generally applicable to turbulent flow without restriction on
dp*/dx*.
Exercise Problem 6.28: Determination of heat transfer rate for
prescribed turbine blade operating conditions from wind tunnel
data obtained for a geometrically similar but smaller blade. The
blade surface area may be assumed to be directly proportional
to
its characteristic length . (1/2)
SCHEMATIC:
Exercise Problem 6.28: Determination of heat transfer rate for
prescribed turbine blade operating conditions from wind tunnel
data obtained for a geometrically similar but smaller blade. The
blade surface area may be assumed to be directly proportional
to
its characteristic length . (2/2)
<
Exercise Problem 6.39: Use of a local Nusselt
number correlation to estimate the surface
temperature of a chip on a circuit board. (1/3)
SCHEMATIC:
Exercise Problem 6.39: Use of a local Nusselt
number correlation to estimate the surface
temperature of a chip on a circuit board. (2/3)
Exercise Problem 6.39: Use of a local Nusselt
number correlation to estimate the surface
temperature of a chip on a circuit board. (3/3)
<
Concentration Boundary Layer
• Features
– A consequence of evaporation or sublimation
of species A from a liquid or solid surface
across which a second fluid species B
is flowing.
– A region of the flow characterized by species
fluxes and concentration gradients.
– A region between the surface and free stream
whose thickness increases in the flow
direction.
– Why does increase in the flow direction?
– Manifested by a surface species flux and a
convection mass transfer coefficient hm.
– What is the heat transfer analog to Fick’s law?
δ c → CA, s − CA ( y ) = 0.99
C
A, s
−
C
A,∞
Definitions (1/2)
Term Variable Units
Species molar flux kmol/s·m
2
Species molar rate kmol/s
Species mass flux kg/s·m
2
Species mass rate kg/s
Species molar concentration kmol/m
3
Species mass concentration (density) kg/m
3
Species molecular weight kg/kmol
Convection mass transfer coefficient m/s
Binary diffusion coefficient1 m
2/s
1 Table A.8
Definitions (2/2)
• Convection Calculations
Species Molar Flux:
Species Mass Flux:
Total Transfer Rates:
Average Mass Transfer Coefficient:
Species Vapor Concentration
or Density (1/2)
• At a Vapor/Liquid or Vapor/Solid Interface
The vapor concentration/density corresponds to saturated
conditions at the
interface temperature Ts .
Assuming perfect gas behavior, the concentration/density may
be estimated
from knowledge of the saturation pressure.
The concentration may also be directly determined from
saturation tables.
For example, from Table A.6 for saturated water,
Species Vapor Concentration
or Density (2/2)
• Free Stream Conditions
– The free stream concentration/density may be determined
from knowledge of the
vapor pressure, assuming perfect gas behavior.
– For water vapor in air, the free stream concentration/density
may be determined
from knowledge of the relative humidity, .
For dry air,
Species Boundary Layer
Equation and Similarity (1/3)
• Species boundary layer approximation:
• Species equation for a non-reacting
boundary layer:
What is the physical significance of each term?
Is this equation analogous to another boundary layer equation?
Species Boundary− Layer
C* ≡ CA CA,s
EquationACA,∞− CandA,s Similarity (2/3)
• Dimensionless form of the species boundary layer equation:
How may the Schmidt number be interpreted?
• Functional dependence for a prescribed geometry:
Species Boundary Layer
Equation and Similarity (3/3)
The dimensionless local convection mass transfer coefficient is
then
What is the functional dependence of the average Sherwood
number?
Analogies (1/2)
• Heat and Mass Transfer Analogy
From analogous forms of the dimensionless boundary layer
energy and species
equations, it follows that, for a prescribed geometry and
equivalent boundary
conditions, the functional dependencies of Nu and Sh are
equivalent.
Since the Pr and Sc dependence of Nu and Sh, respectively, is
typically of the
form Prn and Scn, where n is a positive exponent (0.30 ≤ n ≤
0.40),
Analogies (2/2)
• Reynolds Analogy
• Modified Reynolds Analogy
Colburn j factor for mass transfer
– Applicable to laminar flow if dp*/dx* ~ 0.
– Generally applicable to turbulent flow without restriction on
dp*/dx*.
Evaporative Cooling (1/2)
• The term evaporative cooling originates from association of
the latent energy created
by evaporation at a liquid interface with a reduction in the
thermal energy of the liquid.
If evaporation occurs in the absence of other energy transfer
processes, the thermal
energy, and hence the temperature of the liquid, must decrease.
• If the liquid is to be maintained at a fixed temperature, energy
loss due to evaporation
must be replenished by other means. Assuming convection heat
transfer at the interface
to provide the only means of energy inflow to the liquid, an
energy balance yields
Evaporative Cooling (2/2)
Steady-state
Obtained from heat/mass
Cooling
transfer analogy
• With radiation from the interface and heat addition by other
means,
Exercise Problem 6.64: Use of the naphthalene
sublimation technique to obtain the average convection
heat transfer coefficient for a gas turbine blade. (1/3)
Schematic:
Exercise Problem 6.64: Use of the naphthalene sublimation
technique to obtain the average convection heat transfer
coefficient for a gas turbine blade. (2/3)
Exercise Problem 6.64: Use of the naphthalene sublimation
technique to obtain the average convection heat transfer
coefficient for a gas turbine blade. (3/3)
<
Exercise Problem 6.80: Use of wet and dry bulb
temperature measurements to determine the
relative humidity of an air stream. (1/2)
SCHEMATIC:
(318K): vg = 15.52 m
3/kg; Table A-8, Air-vapor (1 atm, 298 K): D AB= 0.26 ⋅ 10
-4m2/s,
Exercise Problem 6.80: Use of wet and dry bulb
temperature measurements to determine the
relative humidity of an air stream. (2/2)
<
Suggested Problems to Practice
•Example Problem: 6.1 (Page-385) to 6.8 (Page-414)
•Exercise Problem: 6.1 (Page-419) to 6.85 (Page-432)
•Derive equations 6.27, 6.28, and 6.29 showing all the steps to
find
velocity, and temperature distribution for laminar flow inside
the
boundary layer using mass, momentum, and energy
conservations.
41
Homework-4
§Solve all the example problems (6.1 to 6.8) from the text
book from this Chapter-6
§Solve all the exercise problems (6.28, 6.39, 6.64, and
6.80) mentioned in the slides from this Chapter-6
§Show all the steps (Given, Find, Assumptions, Solve,
hand drawings etc.) to give impression that you understood the
problem §Write all the necessary equations applied to those
problems
§Due by Friday 7/27 by 8pm
§Please use the file name for attachment as: 'HW-4-Your First
and Last name' .
42
Please write a response to the discussion below with meaningful
and relevant responses that add value to the discussion.
Joseph Boyd DISCUSSION
Dickenson County Virginia Public Schools has been going
through shrinking pains of a declining budget for many years.
Public school systems typically receive their funding from the
local governments who receive their funding allocation from the
State and Federal levels. Dickenson County Virginia has
experienced a declining student population for many years due
citizens moving from the area further contributing to a
shrinking economy. A major problem the community, the local
governments and the school board system faces is with the
citizen unrest and protests to keep local schools open. Another
co-problem is with the teachers employed by the school system
are protesting for their jobs and making demands in a dire
situation. Citizens, led mostly by teachers and the politicians
attempting to influence their local communities, are holding
rallies, protests and doing all they can to garner support and
show numbers at School Board meetings and local community
activist events. Loosing jobs, closing businesses or schools is
challenging and, in all respect bad for an already weakened
economy. Regardless of what any political leader may say, the
answer, if it only “calms the storm for a short while” will
ultimately surface again when the next budget cycle completes,
usually with more urgency and a vengeance.
The problem faced by Dickenson County Public School systems
transcends problems of a government official communicating
with the populace that elected the official. With the declining
economy in several counties of Southwest Virginia, the State
and Federal governments have invested dollars in attempt to
stimulate an economy that could grow beyond the losses of a
once thriving coal industry. Tourism stimulus, through the
creation of Recreational Authorities to establish entertainment
opportunities, crosses local government boundaries and private
borders and have been in development since 2006. A major
challenge in stimulating this growth has centered on property
owner rights whether from private citizens or from the former
energy companies. Through the established recreation
authority, liabilities that land owners once feared have been
resolved. The tourism development brought with the formation
of the Spearhead Trails (Spearhead Trails Timeline Chart, 2017)
initiative has taken ten years to develop ATV and equestrian
trails and is maturing well enough for businesses to begin
investing and growing in Southwest Virginia.
The Problem
Leaders can unknowingly maintain problems when
communication and linguistics is not within the skills and
abilities of the leader or in this case, the elected official.
Communicating with the citizens in Southwest Virginia is
certainly challenging and carries a stereotypical mindset of the
Appalachians and “Hillbillies” to name a few. In truth,
communication is challenging as is the motivation of individual
citizens to embrace change and accept the results of a changing
economy or embrace the suggested new solutions to the
problem.
Citizen engagement in the development of a tourism economy is
mandatory. According to Herman, V., Deac, L., Ciobotaru, A.
Andronache, C., Loghin, V. & Ilie, M. (2017), tourism “sets up
a significant indicator regarding the development level recorded
by a human collectivity in a certain area.” Land is required,
local business investment, new business startups in travel,
hospitality and boutique shopping become feasible. There are
many stages of tourism development but typically the beginning
is an event or site that attracts tourists followed by lodging and
places to eat and on to other recreational activities. The simple
steps outlined here have yet to be fully embraced by the citizens
in the area, evidenced by tourism inflow but common feedback
being negative due to lack of lodging and “nice” places to eat.
The Problem Redefined
Restating a problem (Block, 2011), is one step to the root cause
of a problem, by educating the citizens that are in a state of
unrest, unwilling to accept change and embrace the new
economy possibilities. While this economy stimulus may not
bring the opportunity for the former times, it will certainly
bring better opportunities than the current economy state.
Bringing a more comprehensive time frame going back to date
when the public schools once operated and thrived in, in
relation to the economy and citizens population may help ease
the unrest while correlating the multiple factors to achieve the
desired community action.
Questions to Ask
Questions could be asked of the local politicians, the school
board officials or of the local citizens. An attempt to stimulate
the economy and gain the citizens buy in to a better economy
could be achieved through enlightenment to the surrounding
facts of a thriving economy and public-school system.
1. What was the state of the economy, population, median
household income and population in Dickenson County
Virginia, when the School system was able to operate in or
nearly operate within the financial allocations based on
population?
2. What type of economy was the leading economic driver for
these times?
3. What steps can be taken to develop or replace the economy
that once existed to create a community where public-school
funding was not a critical daily challenge?
Reactions
When people are in an emotional uprising state, local
government and official statements are not likely going to assist
with achieving any desired state of calm. An outside consultant
study may help gather the facts that can be communicated
properly to the citizens, through proper cultural considerations,
linguistics and creative charting. The answers from the current
school administration and local government officials will likely
be as expected from the questions and their correct answers.
Helping the officials to achieve the results of a motivated
citizenry that will provide an opportunity for a growing
economy is the desired result instead of the current state of
unrest.
Please write a response to the discussion below with meaningful
and relevant responses that add value to the discussion.
Jennifer Lee-Hooper DISCUSSION
A consultant is often in a unique position in which their
external relationship to the organization allows for greater
potential for objectivity than the invested client. A consultant
has this advantage while simultaneously entering the
relationship with a slight disadvantage; they are considered an
outsider. The consultant’s external relationship to the
organization eliminates familiarity. The culture and accepted
norms of the organization may be unfamiliar to the consultant.
The consequences of this vital piece of information can
sometimes influence the proper assessment of the situation. A
consultant, therefore, must rely upon the relationships they
create with their client and the constituents of change.
This assignment leans more toward a scenario of an internal
consultant, with request to provide perceived problems of a
familiar organization. Organizations can face challenges in a
variety of manner, such as financial solvency, image concerns,
as well as fluctuating performance metrics. The Veterans
Affairs administration has faced many of these challenges in
their tenure as an organization. In recent years the VA faced
several image issues due to discovery of ineffective processes
and mismanaged patient paperwork (Schoen, 2017). My
acquaintance with the VA occurred over a decade prior. The
established practices were archaic to the standards of that time.
One major problem was the inefficiency of administrative filing
and documentation standards.
As a potential consultant of the VA, the redefinition of the
problem would be a clarification of the exact processes desired
by the administration to digitize. At the time of employment,
my department was still using carbon copy documentation and
relying upon staff members to hand deliver copies to multiple
departments. The prevalence of available technology amongst
the industry appeared obvious to most staff members, yet
budgetary constraints limited activity to make organizational
changes.
Discovering the client’s focus for solving their problem would
rely upon dialogue between consultant and client in which a
mixture of open-ended and confirmation (close-ended) questions
should be employed. To begin, a basic question such as, “What
are you looking to accomplish?” can be made. This informs the
client that the consultant is engaged in the endeavor to offer
their assistance. The forum for the client to make introductions
into the perceived problem has been established. The
conversation should develop with open-ended questions to elicit
more detail from the client regarding their perceived notion of
the problem in their current situation. As necessary, the
consultant may interject clarifying questions to assure their
accurate reception of the client’s intentions. The precision of
the discovery phase will inform the consultant of the necessary
plan they’ll construct for presentation to the client.
The discovery phase can be a daunting endeavor (J. Williams,
personal communication, July 2018). It takes quite a bit of
invested time to research the organization, learn of its mission
and values as well as its recent trends within its industry. The
consultant must already possess or make themselves aware of
competitors to the organization. Their knowledge of best
practices, accepted within the industry, should also prove useful
as this will create context. The consultant must align themselves
with the organization (J. Williams, personal communication,
July 2018). Additional open-ended questions are outlined
below:
· What is your vision for the organization?
· What are your available resources? (inclusive of budget,
human capital, and physical resources)
· What do you feel are your limitations?
· What is your desired timeframe?
As the consultant gathers data from their client, confirmation
statements can be used to assure the client and consultant are
understanding the same intention. In the case of the VA, a
hypothetical consultant may have communicated with senior
leadership of the administration to determine the necessity for
updating records into a digitized format. Specific departments
may have been selected to triage as a higher priority than others
(such as medical records of active patients prior to updating
records that had existing redundancy measures in place). The
consultant may say, “I just wanted to repeat back a few items
that you’ve mentioned thus far to assure this sounds accurate to
you…” Confirmation and understanding (Argosy University
lecture, 2018) are necessary for the relationship to be successful
as it progresses into the next phases of the consulting cycle.
Lok (2018) states, that consulting is about relationships and
relationships are based on feelings. The consultant’s role should
nurture the relationship (Lok, 2018) from the beginning. The
discovery phase allows the consultant to create an environment
with the client that provides assurance in their plan to move
forward together.
The strength of the relationship built between client and
consultant begins with the investment shown by the consultant
to take the time in getting to know their prospective client. The
consultant does not want to be seen as a mere commodity (Lok,
2018) but a resource to the organization. A good consultant will
assist in providing a viable solution to enhance the organization
(Lok, 2018). The initial outlining of problems and discovery of
the true reality of an organization’s solvency can be shocking.
The client may feel discomfort or surprise to be shown a
glimpse of a problem that was unknown or perhaps avoided.
This narrow perception has been referred to as the inability to
see the forest through the trees. The client may face
disappointment either because of the discovered problems or
perhaps out of fear and unwillingness to make changes.
Responsible and direct communication assures the consultant
can effectively navigate conversations with their clients and
provide reasonable assurance that options may exist. The client
needs to have hope their problems, or challenges, can attempt to
be solved. Once the consultant has conducted a thorough search
and discovered a satisfactory amount of data to make their
conclusion, a pitch is given to gain commitment and proceed
with a contract. It is the consultant’s responsibility to show
investment to the client as well as present the notion that the
client cannot achieve the success they desire without the
assistance of the consultant/consulting firm. The research
(discovery) garnered by the consultant should provide
confidence to the consultant to sell this idea to their client.
The initial stage of discovery allows the consultant to construct
a presentation that should intrigue the client. Their goal is to
gather enough attention to bolster deeper commitment. As
consulting is more relational than transactional (Lok, 2018), the
initial discovery phase may better be suited to occur external to
the confines of structured inquiry. The consultant should be
confident in their skillset, industry knowledge, and research
about the client to host an introductory discovery session.
Further, more measured discovery should occur later in the
cycle, upon collection of additional data and subsequent
analysis. The consultant, once again, may dialogue with the
client to discover more details of their needs to get a better
understanding of the heart of the problem (Argosy University
lecture, 2018) . That information should then be calculated into
the diagnosis and intervention design (Argosy University
lecture, 2018).
The consultant may experience a myriad of emotions
when asked questions that make them conduct self analysis and
reflection. Change and flexibility are essential characteristics of
successful organizations (Buono, 2011). The client’s
willingness to engage with a consultant shows their interest in
maintaining relevance and sustainability. The discovery phase is
a necessary component of the consulting process and step
toward making positive change.

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Fundamentals of Heat and Mass Transfer

  • 1. Fundamentals of Heat and Mass Transfer, Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera, David P. DeWitt, John Wiley & Sons, Inc. •Chapter 1: Introduction Conduction Heat Transfer •Chapter 2: Introduction to Conduction •Chapter 3: 1D, Steady-State Conduction •Chapter 4: 2D, Steady-State Conduction •Chapter 5: Transient Conduction Convection Heat Transfer •Chapter 6: Introduction to Convection •Chapter 7: External Flow •Chapter 8: Internal Flow •Chapter 9: Free Convection •Chapter 10: Boiling and Condensation •Chapter 11: Heat Exchangers Radiation Heat Transfer •Chapter 12: Radiation Processes and Properties •Chapter 13: Radiation Exchange Between Surfaces 1 Mass Transfer
  • 2. •Chapter 14: Diffusion Mass Transfer Chapter-6 (Introduction to Convection) 2 Chapter-6: Introduction to Convection (1/2) 6.1 The Convection Boundary Layers 378
  • 3. 6.1.1 The Velocity Boundary Layer 6.1.2 The Thermal Boundary Layer 6.1.3 The Concentration Boundary Layer 6.1.4 Significance of the Boundary Layers 6.2 Local and Average Convection Coefficients 6.2.1 Heat Transfer 6.2.2 Mass Transfer 6.2.3 The Problem of Convection 6.3 Laminar and Turbulent Flow 6.3.1 Laminar and Turbulent Velocity Boundary Layers 6.3.2 Laminar and Turbulent Thermal and Species Concentration Boundary Layers 3 Chapter-6: Introduction to Convection (2/2) 6.4 The Boundary Layer Equations 6.4.1 Boundary Layer Equations for Laminar Flow 6.4.2 Compressible Flow 6.5 Boundary Layer Similarity: The Normalized Boundary Layer Equations
  • 4. 6.5.1 Boundary Layer Similarity Parameters 6.5.2 Functional Form of the Solution s 6.6 Physical Interpretation of the Dimensionless Parameters 6.7 Boundary Layer Analogies 6.7.1 The Heat and Mass Transfer Analogy 6.7.2 Evaporative Cooling 6.7.3 The Reynolds Analogy 6.8 Summary 4
  • 5. Boundary Layers: Physical Features • Velocity Boundary Layer – A consequence of viscous effects associated with relative motion between a fluid and a surface. – A region of the flow characterized by shear stresses and velocity gradients. – A region between the surface and the free stream whose thickness ( ) increases in the flow direction. – Why does increase in the flow direction?
  • 6. – Manifested by a surface shear stress that provides a drag force (FD). – How does vary in the flow direction? Why? Thermal Boundary Layer – A consequence of heat transfer between the surface and fluid. – A region of the flow characterized by temperature gradients and heat fluxes. – A region between the surface and the free stream whose thickness
  • 7. increases in the flow direction. – Why does increase in the flow direction? – Manifested by a surface heat flux qsʹʹ and a convection heat transfer coefficient h. – If is constant, how doand h vary in the flow direction? Distinction between Local and Average Heat Transfer Coefficients • Local Heat Flux and Coefficient: • Average Heat Flux and Coefficient for a Uniform Surface
  • 8. Temperature: q = ∫As q ʹʹ dAs • For a flat plate in parallel flow: Boundary Layer Transition
  • 9. • How would you characterize conditions in the laminar region of boundary layer development? In the turbulent region? • What conditions are associated with transition from laminar to turbulent flow? • Why is the Reynolds number an appropriate parameter for quantifying transition from laminar to turbulent flow? • Transition criterion for a flat plate in parallel flow: What may be said about transition if ReL<Rex,c? If ReL>Rex,c? • Effect of transition on boundary layer thickness and local
  • 10. convection coefficient: Why does transition provide a significant increase in the boundary layer thickness?
  • 11. Why does the convection coefficient decay in the laminar region? Why does it increase significantly with transition to turbulence, despite the increase in the boundary layer thickness? Why does the convection coefficient decay in the turbulent region? Boundary Layer Equations (1/3)
  • 12. • Consider concurrent velocity and thermal boundary layer development for steady, two-dimensional, incompressible flow with constant fluid properties and negligible body forces. • Apply conservation of mass, Newton’s 2nd Law of Motion and conservation of energy to a differential control volume and invoke the boundary layer approximations. Velocity Boundary Layer:
  • 13. Thermal Boundary Layer: Boundary Layer Equations (2/3) Conservation of Mass: In the context of flow through a differential control volume, what is the physical significance of the foregoing terms, if each is multiplied by the mass density of the fluid? • Newton’s Second Law of Motion:
  • 14. What is the physical significance of each term in the foregoing equation? Why can we express the pressure gradient as dp∞/dx instead of ∂p / ∂x ? Boundary Layer Equations (3/3) Conservation of Energy: What is the physical significance of each term in the foregoing equation? What is the second term on the right-hand side called and under
  • 15. what conditions may it be neglected? Boundary Layer Similarity (1/4) • As applied to the boundary layers, the principle of similarity is based on determining similarity parameters that facilitate application of results obtained for a surface experiencing one set of conditions to geometrically similar surfaces experiencing different conditions. (Recall how introduction of the similarity parameters Bi and Fo permitted generalization of results for transient, one- dimensional conduction). • Dependent boundary layer variables of interest are: • For a prescribed geometry, the corresponding independent
  • 16. variables are: Geometrical: Size (L), Location (x,y) Hydrodynamic: Velocity (V) Fluid Properties: and T = f (x , y , L, V , ρ , µ, c p , k , Ts ,T∞ ) Boundaryh=f(x,L,V,ρ,µ, c Layerp,k,Ts,T∞) Similarity (2/4) • Key similarity parameters may be inferred by non- dimensionalizing the momentum and energy equations. • Recast the boundary layer equations by introducing dimensionless forms of
  • 17. the independent and dependent variables. • Neglecting viscous dissipation, the following normalized forms of the x-momentum and energy equations are obtained: Boundary Layer Similarity (3/4) How may the Reynolds and Prandtl numbers be interpreted physically?
  • 18. • For a prescribed geometry, The dimensionless shear stress, or local friction coefficient, is then What is the functional dependence of the average friction
  • 19. coefficient? Boundary Layer Similarity (4/4) For a prescribed geometry, The dimensionless local convection coefficient is then
  • 20. What is the functional dependence of the average Nusselt number? How does the Nusselt number differ from the Biot number? Reynolds Analogy (1/2) • Equivalence of dimensionless momentum and energy equations for negligible pressure gradient (dp*/dx*~0) and Pr~1:
  • 21. Advection terms Diffusion • Hence, for equivalent boundary conditions, the solutions are of the same form: u * = T * ∂u * = ∂T * ∂y * ∂y* * = 0 * =0 y y C f
  • 22. Re 2 = Nu Reynolds Analogy (2/2) With Pr = 1, the Reynolds analogy, which relates important parameters of the velocity and thermal boundary layers, is • Modified Reynolds (Chilton-Colburn) Analogy: – An empirical result that extends applicability of the Reynolds analogy:
  • 23. Colburn j factor for heat transfer – Applicable to laminar flow if dp*/dx* ~ 0. – Generally applicable to turbulent flow without restriction on dp*/dx*. Exercise Problem 6.28: Determination of heat transfer rate for prescribed turbine blade operating conditions from wind tunnel data obtained for a geometrically similar but smaller blade. The blade surface area may be assumed to be directly proportional to its characteristic length . (1/2)
  • 24. SCHEMATIC: Exercise Problem 6.28: Determination of heat transfer rate for prescribed turbine blade operating conditions from wind tunnel data obtained for a geometrically similar but smaller blade. The blade surface area may be assumed to be directly proportional to its characteristic length . (2/2)
  • 25. < Exercise Problem 6.39: Use of a local Nusselt number correlation to estimate the surface temperature of a chip on a circuit board. (1/3)
  • 27. Exercise Problem 6.39: Use of a local Nusselt number correlation to estimate the surface temperature of a chip on a circuit board. (2/3) Exercise Problem 6.39: Use of a local Nusselt number correlation to estimate the surface temperature of a chip on a circuit board. (3/3) < Concentration Boundary Layer • Features – A consequence of evaporation or sublimation of species A from a liquid or solid surface across which a second fluid species B
  • 28. is flowing. – A region of the flow characterized by species fluxes and concentration gradients. – A region between the surface and free stream whose thickness increases in the flow direction. – Why does increase in the flow direction? – Manifested by a surface species flux and a convection mass transfer coefficient hm. – What is the heat transfer analog to Fick’s law? δ c → CA, s − CA ( y ) = 0.99 C A, s − C
  • 29. A,∞ Definitions (1/2) Term Variable Units Species molar flux kmol/s·m 2 Species molar rate kmol/s Species mass flux kg/s·m 2 Species mass rate kg/s Species molar concentration kmol/m 3
  • 30. Species mass concentration (density) kg/m 3 Species molecular weight kg/kmol Convection mass transfer coefficient m/s Binary diffusion coefficient1 m 2/s 1 Table A.8 Definitions (2/2)
  • 31. • Convection Calculations Species Molar Flux: Species Mass Flux: Total Transfer Rates: Average Mass Transfer Coefficient:
  • 32. Species Vapor Concentration or Density (1/2) • At a Vapor/Liquid or Vapor/Solid Interface The vapor concentration/density corresponds to saturated conditions at the interface temperature Ts . Assuming perfect gas behavior, the concentration/density may be estimated from knowledge of the saturation pressure. The concentration may also be directly determined from saturation tables. For example, from Table A.6 for saturated water, Species Vapor Concentration
  • 33. or Density (2/2) • Free Stream Conditions – The free stream concentration/density may be determined from knowledge of the vapor pressure, assuming perfect gas behavior. – For water vapor in air, the free stream concentration/density may be determined from knowledge of the relative humidity, . For dry air, Species Boundary Layer Equation and Similarity (1/3)
  • 34. • Species boundary layer approximation: • Species equation for a non-reacting boundary layer: What is the physical significance of each term? Is this equation analogous to another boundary layer equation? Species Boundary− Layer C* ≡ CA CA,s EquationACA,∞− CandA,s Similarity (2/3) • Dimensionless form of the species boundary layer equation:
  • 35. How may the Schmidt number be interpreted? • Functional dependence for a prescribed geometry: Species Boundary Layer Equation and Similarity (3/3) The dimensionless local convection mass transfer coefficient is then
  • 36. What is the functional dependence of the average Sherwood number? Analogies (1/2) • Heat and Mass Transfer Analogy From analogous forms of the dimensionless boundary layer energy and species equations, it follows that, for a prescribed geometry and equivalent boundary conditions, the functional dependencies of Nu and Sh are equivalent.
  • 37. Since the Pr and Sc dependence of Nu and Sh, respectively, is typically of the form Prn and Scn, where n is a positive exponent (0.30 ≤ n ≤ 0.40), Analogies (2/2) • Reynolds Analogy
  • 38. • Modified Reynolds Analogy Colburn j factor for mass transfer – Applicable to laminar flow if dp*/dx* ~ 0. – Generally applicable to turbulent flow without restriction on dp*/dx*.
  • 40. • The term evaporative cooling originates from association of the latent energy created by evaporation at a liquid interface with a reduction in the thermal energy of the liquid. If evaporation occurs in the absence of other energy transfer processes, the thermal energy, and hence the temperature of the liquid, must decrease. • If the liquid is to be maintained at a fixed temperature, energy loss due to evaporation must be replenished by other means. Assuming convection heat transfer at the interface to provide the only means of energy inflow to the liquid, an energy balance yields Evaporative Cooling (2/2)
  • 42. transfer analogy • With radiation from the interface and heat addition by other means, Exercise Problem 6.64: Use of the naphthalene sublimation technique to obtain the average convection heat transfer coefficient for a gas turbine blade. (1/3)
  • 43. Schematic: Exercise Problem 6.64: Use of the naphthalene sublimation technique to obtain the average convection heat transfer coefficient for a gas turbine blade. (2/3) Exercise Problem 6.64: Use of the naphthalene sublimation technique to obtain the average convection heat transfer coefficient for a gas turbine blade. (3/3)
  • 44. < Exercise Problem 6.80: Use of wet and dry bulb temperature measurements to determine the relative humidity of an air stream. (1/2) SCHEMATIC:
  • 45. (318K): vg = 15.52 m 3/kg; Table A-8, Air-vapor (1 atm, 298 K): D AB= 0.26 ⋅ 10 -4m2/s, Exercise Problem 6.80: Use of wet and dry bulb temperature measurements to determine the relative humidity of an air stream. (2/2)
  • 46.
  • 47. < Suggested Problems to Practice •Example Problem: 6.1 (Page-385) to 6.8 (Page-414) •Exercise Problem: 6.1 (Page-419) to 6.85 (Page-432) •Derive equations 6.27, 6.28, and 6.29 showing all the steps to find velocity, and temperature distribution for laminar flow inside the boundary layer using mass, momentum, and energy conservations.
  • 48. 41
  • 49. Homework-4 §Solve all the example problems (6.1 to 6.8) from the text book from this Chapter-6 §Solve all the exercise problems (6.28, 6.39, 6.64, and 6.80) mentioned in the slides from this Chapter-6 §Show all the steps (Given, Find, Assumptions, Solve, hand drawings etc.) to give impression that you understood the problem §Write all the necessary equations applied to those problems §Due by Friday 7/27 by 8pm §Please use the file name for attachment as: 'HW-4-Your First and Last name' . 42
  • 50. Please write a response to the discussion below with meaningful and relevant responses that add value to the discussion. Joseph Boyd DISCUSSION Dickenson County Virginia Public Schools has been going through shrinking pains of a declining budget for many years. Public school systems typically receive their funding from the local governments who receive their funding allocation from the State and Federal levels. Dickenson County Virginia has experienced a declining student population for many years due citizens moving from the area further contributing to a shrinking economy. A major problem the community, the local governments and the school board system faces is with the citizen unrest and protests to keep local schools open. Another co-problem is with the teachers employed by the school system are protesting for their jobs and making demands in a dire situation. Citizens, led mostly by teachers and the politicians attempting to influence their local communities, are holding rallies, protests and doing all they can to garner support and show numbers at School Board meetings and local community activist events. Loosing jobs, closing businesses or schools is
  • 51. challenging and, in all respect bad for an already weakened economy. Regardless of what any political leader may say, the answer, if it only “calms the storm for a short while” will ultimately surface again when the next budget cycle completes, usually with more urgency and a vengeance. The problem faced by Dickenson County Public School systems transcends problems of a government official communicating with the populace that elected the official. With the declining economy in several counties of Southwest Virginia, the State and Federal governments have invested dollars in attempt to stimulate an economy that could grow beyond the losses of a once thriving coal industry. Tourism stimulus, through the creation of Recreational Authorities to establish entertainment opportunities, crosses local government boundaries and private borders and have been in development since 2006. A major challenge in stimulating this growth has centered on property owner rights whether from private citizens or from the former energy companies. Through the established recreation authority, liabilities that land owners once feared have been resolved. The tourism development brought with the formation of the Spearhead Trails (Spearhead Trails Timeline Chart, 2017) initiative has taken ten years to develop ATV and equestrian trails and is maturing well enough for businesses to begin investing and growing in Southwest Virginia. The Problem
  • 52. Leaders can unknowingly maintain problems when communication and linguistics is not within the skills and abilities of the leader or in this case, the elected official. Communicating with the citizens in Southwest Virginia is certainly challenging and carries a stereotypical mindset of the Appalachians and “Hillbillies” to name a few. In truth, communication is challenging as is the motivation of individual citizens to embrace change and accept the results of a changing economy or embrace the suggested new solutions to the problem. Citizen engagement in the development of a tourism economy is mandatory. According to Herman, V., Deac, L., Ciobotaru, A. Andronache, C., Loghin, V. & Ilie, M. (2017), tourism “sets up a significant indicator regarding the development level recorded by a human collectivity in a certain area.” Land is required, local business investment, new business startups in travel, hospitality and boutique shopping become feasible. There are many stages of tourism development but typically the beginning is an event or site that attracts tourists followed by lodging and places to eat and on to other recreational activities. The simple steps outlined here have yet to be fully embraced by the citizens in the area, evidenced by tourism inflow but common feedback being negative due to lack of lodging and “nice” places to eat. The Problem Redefined Restating a problem (Block, 2011), is one step to the root cause
  • 53. of a problem, by educating the citizens that are in a state of unrest, unwilling to accept change and embrace the new economy possibilities. While this economy stimulus may not bring the opportunity for the former times, it will certainly bring better opportunities than the current economy state. Bringing a more comprehensive time frame going back to date when the public schools once operated and thrived in, in relation to the economy and citizens population may help ease the unrest while correlating the multiple factors to achieve the desired community action. Questions to Ask Questions could be asked of the local politicians, the school board officials or of the local citizens. An attempt to stimulate the economy and gain the citizens buy in to a better economy could be achieved through enlightenment to the surrounding facts of a thriving economy and public-school system. 1. What was the state of the economy, population, median household income and population in Dickenson County Virginia, when the School system was able to operate in or nearly operate within the financial allocations based on population? 2. What type of economy was the leading economic driver for these times? 3. What steps can be taken to develop or replace the economy that once existed to create a community where public-school
  • 54. funding was not a critical daily challenge? Reactions When people are in an emotional uprising state, local government and official statements are not likely going to assist with achieving any desired state of calm. An outside consultant study may help gather the facts that can be communicated properly to the citizens, through proper cultural considerations, linguistics and creative charting. The answers from the current school administration and local government officials will likely be as expected from the questions and their correct answers. Helping the officials to achieve the results of a motivated citizenry that will provide an opportunity for a growing economy is the desired result instead of the current state of unrest. Please write a response to the discussion below with meaningful and relevant responses that add value to the discussion. Jennifer Lee-Hooper DISCUSSION A consultant is often in a unique position in which their external relationship to the organization allows for greater
  • 55. potential for objectivity than the invested client. A consultant has this advantage while simultaneously entering the relationship with a slight disadvantage; they are considered an outsider. The consultant’s external relationship to the organization eliminates familiarity. The culture and accepted norms of the organization may be unfamiliar to the consultant. The consequences of this vital piece of information can sometimes influence the proper assessment of the situation. A consultant, therefore, must rely upon the relationships they create with their client and the constituents of change. This assignment leans more toward a scenario of an internal consultant, with request to provide perceived problems of a familiar organization. Organizations can face challenges in a variety of manner, such as financial solvency, image concerns, as well as fluctuating performance metrics. The Veterans Affairs administration has faced many of these challenges in their tenure as an organization. In recent years the VA faced several image issues due to discovery of ineffective processes and mismanaged patient paperwork (Schoen, 2017). My acquaintance with the VA occurred over a decade prior. The established practices were archaic to the standards of that time. One major problem was the inefficiency of administrative filing and documentation standards. As a potential consultant of the VA, the redefinition of the problem would be a clarification of the exact processes desired
  • 56. by the administration to digitize. At the time of employment, my department was still using carbon copy documentation and relying upon staff members to hand deliver copies to multiple departments. The prevalence of available technology amongst the industry appeared obvious to most staff members, yet budgetary constraints limited activity to make organizational changes. Discovering the client’s focus for solving their problem would rely upon dialogue between consultant and client in which a mixture of open-ended and confirmation (close-ended) questions should be employed. To begin, a basic question such as, “What are you looking to accomplish?” can be made. This informs the client that the consultant is engaged in the endeavor to offer their assistance. The forum for the client to make introductions into the perceived problem has been established. The conversation should develop with open-ended questions to elicit more detail from the client regarding their perceived notion of the problem in their current situation. As necessary, the consultant may interject clarifying questions to assure their accurate reception of the client’s intentions. The precision of the discovery phase will inform the consultant of the necessary plan they’ll construct for presentation to the client. The discovery phase can be a daunting endeavor (J. Williams, personal communication, July 2018). It takes quite a bit of
  • 57. invested time to research the organization, learn of its mission and values as well as its recent trends within its industry. The consultant must already possess or make themselves aware of competitors to the organization. Their knowledge of best practices, accepted within the industry, should also prove useful as this will create context. The consultant must align themselves with the organization (J. Williams, personal communication, July 2018). Additional open-ended questions are outlined below: · What is your vision for the organization? · What are your available resources? (inclusive of budget, human capital, and physical resources) · What do you feel are your limitations? · What is your desired timeframe? As the consultant gathers data from their client, confirmation statements can be used to assure the client and consultant are understanding the same intention. In the case of the VA, a hypothetical consultant may have communicated with senior leadership of the administration to determine the necessity for updating records into a digitized format. Specific departments may have been selected to triage as a higher priority than others (such as medical records of active patients prior to updating records that had existing redundancy measures in place). The consultant may say, “I just wanted to repeat back a few items
  • 58. that you’ve mentioned thus far to assure this sounds accurate to you…” Confirmation and understanding (Argosy University lecture, 2018) are necessary for the relationship to be successful as it progresses into the next phases of the consulting cycle. Lok (2018) states, that consulting is about relationships and relationships are based on feelings. The consultant’s role should nurture the relationship (Lok, 2018) from the beginning. The discovery phase allows the consultant to create an environment with the client that provides assurance in their plan to move forward together. The strength of the relationship built between client and consultant begins with the investment shown by the consultant to take the time in getting to know their prospective client. The consultant does not want to be seen as a mere commodity (Lok, 2018) but a resource to the organization. A good consultant will assist in providing a viable solution to enhance the organization (Lok, 2018). The initial outlining of problems and discovery of the true reality of an organization’s solvency can be shocking. The client may feel discomfort or surprise to be shown a glimpse of a problem that was unknown or perhaps avoided. This narrow perception has been referred to as the inability to see the forest through the trees. The client may face disappointment either because of the discovered problems or perhaps out of fear and unwillingness to make changes. Responsible and direct communication assures the consultant
  • 59. can effectively navigate conversations with their clients and provide reasonable assurance that options may exist. The client needs to have hope their problems, or challenges, can attempt to be solved. Once the consultant has conducted a thorough search and discovered a satisfactory amount of data to make their conclusion, a pitch is given to gain commitment and proceed with a contract. It is the consultant’s responsibility to show investment to the client as well as present the notion that the client cannot achieve the success they desire without the assistance of the consultant/consulting firm. The research (discovery) garnered by the consultant should provide confidence to the consultant to sell this idea to their client. The initial stage of discovery allows the consultant to construct a presentation that should intrigue the client. Their goal is to gather enough attention to bolster deeper commitment. As consulting is more relational than transactional (Lok, 2018), the initial discovery phase may better be suited to occur external to the confines of structured inquiry. The consultant should be confident in their skillset, industry knowledge, and research about the client to host an introductory discovery session. Further, more measured discovery should occur later in the cycle, upon collection of additional data and subsequent analysis. The consultant, once again, may dialogue with the client to discover more details of their needs to get a better understanding of the heart of the problem (Argosy University
  • 60. lecture, 2018) . That information should then be calculated into the diagnosis and intervention design (Argosy University lecture, 2018). The consultant may experience a myriad of emotions when asked questions that make them conduct self analysis and reflection. Change and flexibility are essential characteristics of successful organizations (Buono, 2011). The client’s willingness to engage with a consultant shows their interest in maintaining relevance and sustainability. The discovery phase is a necessary component of the consulting process and step toward making positive change.