Final exam equation sheet
?!"#$ = −??
??
??
, ?!"#% = ℎ?(?& − ?'), ?()$ = ???(?*+ − ?&,(+ )
?(?) = 0?*,. − ?*,/1
?
?
+ ?*,/, 0 < ? < ?
?(?) =
�̇�?.
2?
81 −
?.
?.
: +
?*,. − ?*,
2
?
?
+
?*,/ + ?*,.
2
: − ? < ? < ? plane wall + generation
?(?) =
�̇�?.
2?
81 −
?.
?.
: + ?*: 0 < ? < ? plane wall + generation + one side insulated
?(?) =
?*,/ − ?*,.
ln (?
?.)
ln @
?
?.
A + ?*,., ?/ < ? < ?. radial hollow cylinder
?0 =
2???0?*,/ − ?*,.1
ln (?./?/)
, ?/ < ? < ?. radial hollow cylinder
?(?) =
�̇�?1.
4?
81 −
?.
?1.
: + ?*, 0 < ? < ?1 radial solid cylinder + generation
?
?2
=
? − ?'
?2 − ?'
= exp@−
?
?3
A = exp(−?? ⋅ ??) , loss by convection
?3 = @
?∀?
ℎ?*
A , ?? =
ℎ?4
?
, ?? =
??
?4.
? − ?' − (?/?)
?2 − ?' − (?/?)
= exp (−??), ? = 1/?3 , ? =
?567
?∀?
loss by convection + gain by generation
?(?, ?) − ?*
?2 − ?*
= erf @
?
√4??
A , ?*88(?) =
?(?* − ?2)
√???
, 0 < ? < ∞
?
??
erf @
?
√4??
A =
1
√???
exp8−
?.
4??
:
Criterion for semi − infinite domain:
?(?, ?) − ?*
?2 − ?*
≥ 0.9
?∗ =
?
?2
=
? − ?'
?2 − ?'
= d
'
7:
?7 exp(−?7.??) cos i?7
?
?
j , 0 < ? < ?, ?88(0, ?) = 0
Material constants will be provided if needed. Charts and graphs will be provided if needed.
Heat Transfer – Mass Transfer Analogy
?! "" = −?
∂?
∂?
(
#$%
= ℎ(?! − ?&)
?'!
"" = −?'(
∂?'
∂?
(
#$%
= ℎ)/?!,' − ?&,+0
?!, evap "" = ?+,!"" ℎ,-
??
??./0
=
?ℎ
??./0
?? =
ℎ?
?
?ℎ =
ℎ)?
?+1
?? =
?
?
?? =
?
?+1
ℎ
ℎ)
=
?
?+1
:
?+1
? ;
./0
Flat Plate Boundary Layer
Laminar ??2 < 5 × 103
??2 = 0.332??2
./4??./0 (Eq. 7.23)
??JJJJ2 = 0.664??2
./4??./0 (Eq. 7.30)
Tu
ulent
??2 = 0.0296??2
5/3??./0 (Eq. 7.36)
??JJJJ6 = N0.037??6
5/3 − ?P??./0 (Eq. 7.38)
? = 0.037??2,7
5/3 − 0.664??2,7
./4 (Eq. 7.39)
Pipe Flow
?!"" = ℎ(?! − ?)) (Eq. 8.27)
? = �̇�?8/?),9 − ?),:0 (Eq. 8.34)
?) = ?),: +
?!""?
�̇�?8
? , ?!"" = constant (Eq. 8.40)
?! − ?)(?)
?! − ?),:
= expZ−
??
�̇�?8
ℎ‾\ , ?! = constant (Eq. 8.42)
Δ?9
Δ?:
=
?& − ?),9
?& − ?),:
= exp Z−
?_?!
�̇�?8
\ (Eq. 8.45?)
Cylinder in Cross Flow (Eq. 7.54)
??JJJJ; = 0.3 +
0.62??;
./4 ??./0
[ XXXXXXXXXX/??)4/0]./5
d1 + :
??;
282,000;
3
e
5/3
Flow Through Bank of Tubes
??JJJJ; = ?.??;,=>?) ??%.0A :
??
??B
;
./5
(Eq. 7.58)
??JJJJ;|(D"E4%) = ?4??JJJJ;|(D"G4%) (Eq. 7.59)
?1 and ? from Table 7.5, ?2 from Table 7.6
Table 7.6 will be provided when required
?=>? =
?H
?H − ?
? (Eq. 7.60)
?! − ?9
?! − ?:
= exp Z−
???ℎ‾
???H?H?8
\ (Eq. 7.63)
Δ?I= =
(?! − ?:) − (?! − ?9)
ln N?! − ?:?! − ?9
P
(Eq. 7.62)
?" = ?/ℎ‾??Δ?I=0 (Eq. 7.64)
Pipe Flow
Laminar Flow (??; < 3000) – Thermal Entry
??JJJJ; = 3.66 +
0.0668??;
XXXXXXXXXX??;
4/0 (Eq. 8.57)
1/??; = ?/(???;??) < 0.05
?/(???;) > 0.05
Laminar Flow – Combined Entry
??JJJJ; =
A XXXXXXXXXX??;tanh (??;J.)
tanh N2.432??./A??;
J./AP
(Eq. 8.58)
? =
3.66
tanh s2.264??;
J./0 + 1.7??;
J4/0t
?/(???;??) < 0.05
?/(???;) < 0.05
Tu
ulent Flow
??; = 0.023??;
5/3??%.5 (Eq XXXXXXXXXXHeating)
??; = 0.023??;
5/3??%.0 (Eq XXXXXXXXXXCooling)
?/? > 10
??; =
4�̇�
???
??2 =
??&?
?
??2 =
?&?
?
??;,=>? =
?=>??
?
ℎ = −
???/??|#$%
?! − ?&
?? =
??∗
??∗
(
#∗$%
?
∗(?∗) =
? − ?!
?& − ?!
? =
5
x?&/??
=
5?
x??2
?L ≈
5?
??2
./4??⬚
./0
?
?L
= ??./0 ? = z
+!
?""??! ℎJ =
?
?!(?! − ?&)
ℎ‾ =
1
?
z
6
%
ℎ?? =
1
?
dz
2$
%
ℎNO)?? + z
6
2$
ℎtu
??e
ℎ) =
−?+1? ?+ ??|#$%⁄
?+,! − ?+,&
?ℎ =
??+∗
??∗
(
#∗$%
?/?7 = ??./0
?L
?7
= ??./0
Convection
?? =
??
??
? = ℎ?!??N)
? = ?_?!Δ?I=
? =
??N)
?L9L
Δ?I= =
Δ?9 − Δ?:
ln(Δ?9/Δ?:)
?
??
d
?!(?) − ?(?, ?)
?!(?) − ?)(?)
e
PQ,L
= 0
Fully developed thermal (fd,t)
??
??
(
PQ,L
= =
??!
??
(
PQ,L
Constant Surface
Heat Flux (fd,t)
??)
??
(
PQ,L
Constant Surface
Temperature (fd,t)
??
??
(
,R,L
=
(?! − ?)
(?! − ?))
??)
??
,R,L
??; = 4.36
??; = 3.66
? = ??Δ?!"
Δ?!" =
Δ?# − Δ?$
ln Δ ⁄?# Δ?$
Δ?$ = ?%,' − ?(,'
Δ?# = ?%,) − ?(,)
Δ?$ = ?%,' − ?(,)
Δ?# = ?%,) − ?(,'
? = �̇�%?*,% ?%,' − ?%,) = ?%Δ?%
? = �̇�(?*,( ?(,) − ?(,' = ?(??(
Heat Exchangers
? =
?
?"+,
?"+, = ?"-.Δ?"+,
Parallel flow
Counter flow
NTU =
??
?"-.
? = ?
??
?"-.
,
?"-.
?"+,
? = 5.67×10/0W/m# ⋅ K1
Radiation
?'2 =
?' − ?2
?'?'2
$
Reciprocity relation
?'?'2 = ?2?2'
Summation rule for enclosures
Composite surfaces
E
23$
4
?'2 = 1
?(2)' =
1
∑73$8 ?7
E
73$
8
?7?7'
?' =
?9' − ?'
⁄1 − ?' ?' ?'
=E
23$
4
?' − ?2
?'?'2
$
?' =
?9' − ?'
⁄1 − ?' ?' ?'
=E
23$
4
?'2
?'(2) = E
73$
8
?'7
?"-. = min(?(, ?%)
?":; = max(?(, ?%)
? = ??9 = ???1
?9 = ??1
Radiation exchange
Blackbody
? = ?9 = ??1
? = 1
?$# =
? ?$1 − ?#1
1 − ?$
?$?$
+ 1?$?$#
+ 1 − ?#?#?#
Two surface enclosure
Charts and graphs will be provided if needed. All material constants will be provided.
Free Convection
??; =
?? ?< − ?= ?
??
Vertical Plate
??? = 0.825 +
0.387???
$/A
XXXXXXXXXX/??)B/$A 0/#C
#
Hot upper or cold lower horizontal surface
??? = 0.54???
$/1 , 101 ≤ ?? ≤ 10C, ?? ≥ 0.7
??? = 0.15???
$
, 10C ≤ ?? ≤ 10$$, all ??
Hot lower or cold upper horizontal surface
??? = 0.52???
$/D, 101 ≤ ?? ≤ 10B, ?? ≥ 0.7
Horizontal Cavity
Critical Rayleigh number, ??( = 1708
??? =
‾ℎ?
? = 0.069???
$
??E.EC1
Vertical Cavity
??! = 0.22
P
0.2 + Pr??!
".$% ?
?
&'/)
2 ≤
?
? ≤ 10
??! = 0.18
??
0.2 + ????!
".$*
1 ≤
?
? ≤ 2
??! = 0.42??!
'/)Pr"."'$
?
?
&".+
10 ≤
?
? ≤ 40
?$# =
?$? ?$1 − ?#1
1
?$
+ 1?#
+
1 − ?>,$
?>,$
+
1 − ?>,#
?>,#
Two surface enclosure with radiation shield
1/?? = 1/ℎ(?( + ?G + 1/ℎ%?% Eq. 11.1a
List of tables and graphs that will be provide in the exam if needed
Heat exchangers
Figure 11.10
Parallel flow
Figure 11.11
Counter flow
Figure 11.12
Shell and tube (1 shell)
Figure 11.13
Shell and tube (2 shells)
Figure 11.14
Cross-flow - unmixed
Figure 11.15
Cross-flow – mixed/unmixed
Radiation exchange
Table 13.1
a) Parallel plates with midlines connected by
perpendicula
) Inclined parallel plates of equal width and a
common edge
c) Perpendicular Plates with a Common Edge
Figure 13.4
View factor for aligned parallel rectangles.
Figure 13.5
View factor for coaxial parallel disks.
Figure 13.6
View factor for perpendicular rectangles with a
common edge.
Conduction
Table 5.1
Coefficients used in the one-term approximation to the series
solutions for transient one-dimensional conduction
Appendix B.2
Gaussian E
or Function
Convection
Table 7.5
Constants of Equation 7.58 for the tube bank in cross flow
Table 7.6
Co
ecWon factor ?# of EquaWon 7.59 for ?? < 20
Figure 8.10 (a)
Local Nusselt numbers.
Final exam review
Transport Processes
ENME332 Fall 2023
Final Exam Review
• 120 mins. In class
• Upload to canvas assignments (10 mins) – submit hardcopy
• Chapters 5, 7, 8, 9, 11, 13
• Closed Books, Closed Notes
• Equation Sheet Provided
Material reviewed in this review today is not be exhaustive of all topics covered on the final exam
Topics – Transient Conduction
5.1 The Lumped Capacitance Method
5.2 Validity of the Lumped Capacitance Method
5.3 General Lumped Capacitance Analysis
5.3.1 Radiation Only
5.3.2 Negligible Radiation (convection + generation)
5.5 The Plane Wall with Convection
5.5.1 Exact Solution
5.5.2 Approximate Solution
5.7 The Semi-Infinite Solid
Topics – Boundary Layer Flow
7.1 The Empirical Method
7.2.1 Laminar Flow over an Isothermal Plate: A Similarity Solution
7.2.2 Tu
ulent Flow over an Isothermal Plate
7.2.3 Mixed Boundary Layer Conditions
7.4.2 Convection Heat and Mass Transfe
8.1.1 Flow Conditions
8.1.2 The Mean Velocity
8.2 Thermal Considerations
8.3 The Energy Balance
8.4 Laminar Flow in Circular Tubes: Thermal Analysis and
Convection Co
elations
8.5 Convection Co
elations: Tu
ulent Flow in Circular Tubes
Topics – Pipe Flow
Topics – Free Convection
9.1 Physical Considerations
9.2 The Governing Equations for Laminar Boundary Layers
9.3 Similarity Considerations
9.4 Laminar Free Convection on a Vertical Surface
9.5 The Effects of Tu
ulence
9.6 Empirical Co
elations: External Free Convection Flows
9.6.1 The Vertical Plate
9.6.2 Inclined and Horizontal Plates
9.7 Free Convection Within Parallel Plate Channels
9.7.1 Vertical Channels
9.8 Empirical Co
elations: Enclosures
9.8.1 Rectangular Cavities
Topics – Heat Exchangers
11.1 Heat Exchanger Types
11.2 The Overall Heat Transfer Coefficient
11.3 Heat Exchanger Analysis: Use of the Log Mean Temperature
Difference
11.4 Heat Exchanger Analysis: The Effectiveness–NTU Method
11.5 Heat Exchanger Design and Performance Calculations
13.1 The View Facto
13.1.1 The View Factor Integral
13.1.2 View Factor Relations
13.2 Blackbody Radiation Exchange
13.3 Radiation Exchange Between Opaque, Diffuse, Gray Surfaces in
an Enclosure
13.3.1 Net Radiation Exchange at a Surface
13.3.2 Radiation Exchange Between Surfaces
13.3.3 The Two-Surface Enclosure
13.3.4 Two-Surface Enclosures in Series and Radiation Shields
13.3.5 The Reradiating Surface
13.4 Multimode Heat Transfe
Topics – Radiation
Free Convection
??
??
0
??
??
0
??
??
0
??
??
0
Vertical Surface
??! =
?? ?" − ?# ?$
?%
Grashoff Numbe
Measures the relative strength of
uoyancy and viscous forces
Rayleigh Numbe
??! = ??!?? =
?? ?" − ?# ?$
??
Factors in thermal diffusion in the
elative strength of buoyancy and
viscous forces
??! = ? ??! , ?? = ?(??!)
Nusselt Numbe
??! = 0.825 +
0.387??!
"/$
XXXXXXXXXX/??)%/"$ &/'(
'
(Eq. 9.26)
?& = ?? ?" − ?# ?
Horizontal Surface
??! = 0.52??!
'/) (Eq. 9.32)
??! = 0.54??!
'/* (Eq. 9.30)
Upper Surface of Cold Plate
Upper Surface of Hot Plate Lower Surface of Cold