Heat and Mass Transfer CHEE330 β Formula Sheet
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Fourierβs law
π"οΏ½οΏ½οΏ½ =ππ΄
= βπππ π Heat transfer rate through area A [W] π"οΏ½οΏ½οΏ½ = π
π΄ Heat flux or heat transfer rate per unit area perpendicular to the
transport direction [W/m2=J/(s m2] π΄ Area perpendicular to heat flux [m2] k Thermal conductivity [W/(m K)] βπ Temperature gradient (driving force) [K/m] 1-Dimensional Fourierβs law for different coordinate systems Fourierβs law expressions and solutions for heat fluxes, heat rates and thermal resistances for steady-state, 1D heat transfer, constant k in various coordinate systems
Plane Wall (Cartesian) Cylindrical Wall Spherical Wall
Fourierβs law π"π₯ = βπππππ
π"π = βπππππ
π"π = βπππππ
Heat flux π" πβππΏ
πβπ
πππ οΏ½π2π1οΏ½
πβπ
π2 οΏ½1π1β 1π2οΏ½
Heat transfer rate π ππ΄
βππΏ
2ππΏπβπ
ππ οΏ½π2π1οΏ½
4ππβπ
οΏ½1π1β 1π2οΏ½
Thermal resistance
Rcond #
πΏππ΄
ππ οΏ½π2π1
οΏ½
2ππΏπ
οΏ½1π1β 1π2οΏ½
4ππ
#Ar=2ΟrL for cylindrical, Ar=4Οr2 for spherical coordinates, r1=rin, r2=rout Radiation Stefan-Boltzmann law for an ideal radiator (black body)
πππ" = πΈ = πππ 4 πππ" = radiation/heat flux emitted from the surface Ts = absolute temperature of the surface [K] Ο = Stefan-Boltzmann constant
For a real (non-ideal) surface πππ" = πΈ = ππππ 4
Ξ΅= emissivity [-] -> black bodies: π=1, real surface: 0<π<1 Irradiation
ππππ" = πΊ = πΌ π ππ π π4 G = rate of incident radiation per unit area (W/m2) of the surface (radiation/heat flux absorbed by the surface) originating from its surroundings Tsur = absolute temperature of the surroundings [K]
Ξ± = absorptivity of the surface [0< Ξ± <1], for a βgreyβ surface Ξ±=Ξ΅
Net radiation exchange ππππ" = πΈ β πΊ = ππππ 4 β πΌ π ππ π π4 = βπ(ππ β ππ π π)
Radiative heat transfer coefficient for grey surface βπ = ππ(ππ + ππ π π)(ππ 2 β ππ π π2 ) [W/m2K]
Thermal circuits
π =πππππππ πππππππ πππππ
π ππ ππ π ππππ=βππ
R = thermal resistance [K/W] Conductive resistance
Rcond = depends on geometry, see table left Convective resistance
π ππππ =1β π΄
Radiative resistance
π πππ =1
βπ π΄
Thermal contact resistance
π "π‘,π =ππ΄ β ππ΅ππ₯"
TA,B = temperature contact surface A,B [K]
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Resistance in series (q=const):
π π‘ππ‘ = π 1 + π 2+. . +π π = οΏ½π ππ
Resistances in parallel (ΞT =const): 1π π‘ππ‘
=1π 1
+1π 2
+. . +1π π
= οΏ½1/π ππ
Ideal gas law
π π = π π π =πππ π
π = pressure [Pa] V = volume [m3] n = molar amount of substance [mol] m = mass of substance [kg] M = Molar mass of substance [mol/g] T = Temperature in K [K] R = universal gas constant = 8.3143 J/(mol K)
Buckingham method: Step 1: List all independent variables involved in the problem Q0 = F(Q1, Q2, ... , Qn) Step 2: Express each of the variables in terms of basic dimensions Step 3: Apply Buckingham π± theorem / Determine number of π± groups: Number of dimensionless groups required to describe the problem is k=(n+1)-j. n = number of independent variables identified for the problem j = number of primary dimensions which have been used to express the variables. Step 4: Selection of a dimensionally independent subset of (repeating) j variables Q1...Qj (j β€ n). Step 5: Build π± groups by multiply one of the nonrepeating variables by the product of the repeating variables, each raised to an exponent that will make the combination dimensionless. Step 6: Assume dimensional homogeneity and solve set of equations to obtain π± groups Step 7: Express result in form π±1 = πΉ(π±2,π±3. .π±π)
Concentrations in a binary system of A and B
Assumptions: ideal Gas
Diffusive molar and mass fluxes for binary system A in B Diffusive Flux Vector notation 1D planar (Cartesian) Molar flux (Fickβs Law)
π ππ π = ππππ π π½π΄οΏ½οΏ½οΏ½ = βπ·π΄π΅πππ΄ π½π΄,π§ = βπ·π΄π΅πππ΄ππ
Mass flux (Fickβs Law) π ππ π = ππππ π π₯π΄οΏ½οΏ½οΏ½ = βπ·π΄π΅πππ΄ ππ΄,π§ = βπ·π΄π΅
πππ΄ππ
Molar flux (de Groot) π½π΄οΏ½οΏ½οΏ½ = βππ·π΄π΅ππ¦π΄ π½π΄,π§ = βππ·π΄π΅ππ¦π΄ππ
Mass flux (de Groot) π₯π΄οΏ½οΏ½οΏ½ = βππ·π΄π΅πππ΄ ππ΄,π§ = βππ·π΄π΅πππ΄ππ
molar flux [mol/(m2s)] mass flux [ kg/(m2s)]
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Absolute molar and mass fluxes for binary system A in B Absolute Flux Vector notation 1D planar (Cartesian)
Molar flux ππ΄οΏ½οΏ½οΏ½οΏ½ = βπ·π΄π΅πππ¦π΄ +π¦π΄(ππ΄οΏ½οΏ½οΏ½οΏ½ + ππ΅οΏ½οΏ½οΏ½οΏ½οΏ½ )
ππ΄,π§ = βπ·π΄π΅πππ¦π΄ππ
+π¦π΄(ππ΄,π§ + ππ΅,π§)
Mass flux ππ΄οΏ½οΏ½οΏ½οΏ½ = βππ·π΄π΅πππ΄ +ππ΄(ππ΄οΏ½οΏ½οΏ½οΏ½ + ππ΅οΏ½οΏ½οΏ½οΏ½ )
ππ΄,π§ = βππ·π΄π΅πππ΄ππ
+ππ΄(ππ΄,π§ + ππ΅,π§) Molar flux for
equimolar counter diffusion
(ππ΄οΏ½οΏ½οΏ½οΏ½ = βππ΅οΏ½οΏ½οΏ½οΏ½οΏ½ )
ππ΄οΏ½οΏ½οΏ½οΏ½ = β DABπππ¦π΄ ππ΄,π§ = DABποΏ½yA,1 β yA,2οΏ½
π2 β π1
Molar flux for unimolecular
diffusion stagnant film (ππ΅οΏ½οΏ½οΏ½οΏ½οΏ½ = 0)
ππ΄οΏ½οΏ½οΏ½οΏ½ = βπ·π΄π΅π
1 β π¦π΄ππ¦π΄
ππ΄,π§ = π·π΄π΅π
(π2 β π1) β
lnοΏ½1 β π¦π΄,2
1 β π¦π΄,1οΏ½
Control volume balance on rate basis In a defined control volume, there is
ACCUMULATION = INPUT - OUTPUT + GENERATION Energy:
ππΈπ ππ‘
= οΏ½οΏ½ππ- οΏ½οΏ½ππ π‘ + οΏ½οΏ½π πΈπ = stored energy [J] οΏ½οΏ½ππ= ingoing energy rate [W] οΏ½οΏ½ππ π‘= outgoing energy rate [W] οΏ½οΏ½π= generated energy rate [W]
Mass Species A πππ΄
ππ = οΏ½οΏ½π΄,ππ β οΏ½οΏ½π΄,ππ π‘ + οΏ½οΏ½π΄,π
ππ΄ = stored mass of A [kg] οΏ½οΏ½π΄,ππ = ingoing mass rate of A [kg/s] οΏ½οΏ½π΄,ππ π‘ = outgoing mass rate of A [kg/s] οΏ½οΏ½π΄,π = generated mass rate of A [kg/s]
Continious flow system
οΏ½οΏ½οΏ½πππ‘ + οΏ½οΏ½πππ‘οΏ½ = οΏ½οΏ½ οΏ½(β2 β β1) +12
(π22 β π12) + π(π2 β π1)οΏ½ οΏ½οΏ½πππ‘ = net heat rate added to CV [W] οΏ½οΏ½πππ‘= net rate of work done in CV [W] οΏ½οΏ½ = mass flow rate [kg/s] ππ = height [m] ππ= velocity [m/s] βπ = ππ ππ = specific enthalpy [J/(kg K)] 1 = inlet, 2=outlet
Differential Equations of Heat Transfer for k = π(ποΏ½οΏ½ )
πππππππ
= π β (πππ ) + οΏ½οΏ½π
for k = constant
ππππ
= πΌβπ +οΏ½οΏ½ππππ
οΏ½οΏ½π= volumetric generation term [W/m3] π = density [kg/m3] Ξ± = k
Ο cp = thermal diffusivity [m2/s] ππ= specific heat capacity [kJ/(kg K)]
Boundary condition of first kind - Dirichlet condition Constant Temperature
π(π = π0, π ) = ππππ π . Boundary condition of second kind - Neumann condition Constant gradient at a boundary (=constant flux)
πππποΏ½π₯=π₯0
= ππππ π .
Boundary condition of third kind - Robin boundary condition The gradient at a boundary is described with a function (e.g. Newtonβs Law of cooling)
πππποΏ½π₯=π₯0
= π(π)
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Differential Equations of Mass Transfer πππ΄ππ
= βπ β ποΏ½οΏ½π΄ + π π΄
ποΏ½οΏ½π΄ can be either the purely diffusive flux π½π΄οΏ½οΏ½οΏ½ or absolute flux ποΏ½οΏ½π΄ of A π π΄ = volumetric rate of mass generation [mol/(s m3)]
πππ΄ππ
= π β (π·π΄π΅πππ¦π΄) β π β οΏ½ππ΄ποΏ½ οΏ½ + π π΄
ποΏ½ =molar-average velocity [m/s] Boundary condition of first kind - Dirichlet condition Constant Temperature
ππ΄(π = π0, π ) = ππππ π . Boundary condition of second kind - Neumann condition Constant gradient at a boundary (=constant flux)
πππ΄ππ
οΏ½π₯=π₯0
= ππππ π .
Boundary condition of third kind - Robin boundary condition The gradient at a boundary is described with a function
πππ΄ππ
οΏ½π₯=π₯0
= π(ππ΄)
Other Boundary conditions for mass transfer Evaporation and sublimation (Raoultβs Law)
ππ΄,π = ππ΄ ππ΄,π ππ‘ ππ΄,π = π¦π΄,π π= partial pressure of A in gas at the surface [bar]
ππ΄,π ππ‘ = saturation (vapor) pressure at the surface Solubility of gases in liquids (Henryβs Law)
ππ΄ = π»ππ΄ π»= Henry constant [Pa] Solubility of gases in solids
ππ΄,π ππ ππ = π ππ΄ π= solubility [Pa m3/mol]
Vector operators for different coordinate systems (f = scalar function, e.g. Temperature T or concentration c):
Vector operators
Cartesian (x,y,z)
Cylindrical (π,π½, π)
Spherical (π,π½,π)
Gradient π΅π
β
ββββ
ππππππππ¦ππππβ
ββββ
β
βββ
ππππ
1ππππΞΈππππ β
βββ
β
ββββ
ππππ
1πππππ
1π π ππ (π)
ππππβ
ββββ
Laplace π΅ππ = π«π
οΏ½π2πππ2
+π2πππ¦2
+π2πππ2
οΏ½
οΏ½1π
ππποΏ½ππππποΏ½
+1π2π2πππ2
+π2πππ2
οΏ½
οΏ½1π2
ππποΏ½π2
πππποΏ½
+1
π2π ππ (π)πππ
οΏ½π ππ (π)πππποΏ½
+1
π2 π ππ2(π) π2πππ2οΏ½
Divergence π΅ β ποΏ½οΏ½
οΏ½ππΉπ₯ππ
+ππΉπ¦ππ
+ππΉπ§ππ
οΏ½
οΏ½1ππ(π πΉπ)ππ
+1πππΉΞΈππ
+ππΉπ§ππ
οΏ½
οΏ½1π2π(π2πΉπ)ππ
+1
π π ππ (π)π(πΉπ π ππ (π))
ππ
+1
π π ππ (π) ππΉπππ
οΏ½
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Convective heat transfer Newtonβs law of Cooling
ππ΄
= π" = ββπ h = convective HT coefficient [W/(m2K)] βπ = temperature difference [K] Internal Flow
πππππ = οΏ½οΏ½πποΏ½ππ,π β ππ,ποΏ½ = β π΄ βππ π Logarithmic temperature difference
βππ π =βππ β βππ
ππ οΏ½βππβπποΏ½
Constant surface temperature βππ = ππ β πππ π‘ βππ = ππ β πππ
Energy balance results in
ππ οΏ½ππ,ππ π‘ β ππ ππ,ππ β ππ
οΏ½ +β
πππππππ4πΏπ·
= 0
Constant external temperature use modified Newtonβs Law
q =βππ ππ π‘ππ‘
π π‘ππ‘ = total resistance of convective and conductive HT βππ π built with
βππ = πβ β πππ π‘ βππ = πβ β πππ
Constant heat flux: Local mean temperature of the fluid:
ππ(π) = ππ,π +ππ "ποΏ½οΏ½ππ
π
Tm,i= mean temperature inlet [K] P=cross section perimeter [m]
m= mass flow rate [kg/s] cp = specific heat capacity [kJ/(kg K)] qsβ= heat flux at the surface [W/m2] Average heat coefficient
βπΏ =1πΏοΏ½ βπ₯πππΏ
0
βπΏ= average heat transfer coefficient over a spatial dimension L βπ₯= local heat transfer coefficient at a certain position x Convective mass transfer
ππ΄ = ππβππ΄ NA = molar convective mass transfer flux [mol/(m2s] ππ= concective mass transfer coefficient [m/s] βππ΄= concentration difference [mol/m3] Internal Flow Use an analogy to HT
Analogy between Heat, Mass and Momentum Transport Skin friction Use local skin friction for analogy of local coefficients
πΆπ,π₯ =2ππ,π₯
ππβ2
ππ,π₯ = Local shear stress at position x [N/m2] Use average skin friction for analogy of average coefficients
πΆπ,πΏ =2ππ,πΏ
ππβ2
ππ,πΏ = πΉπ΄
= Average shear stress = Drag force per surface area over spatial dimension L [N/m2] Reynolds analogy
ππ =β
ππβππ=πΆπ2
= ππ π =πππβ
valid for Blasius solution (laminar flow) of the horizontal plate and Pr=1 and Sc=1
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local skin friction
πΆπ,π₯ =0.664
οΏ½π ππ₯
average skin friction for averaged coefficients
πΆπ,πΏ =1.328
οΏ½π ππΏ
Chilton-Colburn analogy For laminar and turbulent flow where is no form drag such as flow over flat plate and internal flows
ππ» = ππ· =πΆπ2
ππ» = ππ ππ2/3
valid for 0.5<Pr<50
ππ· =πππβ
ππ2/3
valid for 0.6<Sc<2500 Prandtl analogy For turbulent flows where is no form drag such as flow over flat plate and internal flows
ππ =πΆπ/2
1 + 5οΏ½πΆπ 2β (ππ β 1)
for mass transfer Stanton number use Sc instead of Pr. Constants g = Gravitational acceleration =9.81 m2/s kB= Boltzmann constant =1.38 Γ 10-23J/K R = Universal gas constant = 8.3143 J/(mol K)
Ο = Stefan-Boltzmann constant Ο = 5.67x10-8 W/(m2K4) NA = Avogadro number 6.022 Γ 1023 molβ1
Units of selected physical quantities: [Pressure] β‘ atm (standard) = 101325 Pa bar = 105 Pa Pa = N/m2 [Force] β‘ N = kg m/s2 [Work] β‘ J = N m [Power] β‘ W = J/s [Charge] β‘ C [Current] β‘ A = C/s [Voltage] β‘ V = J/C [Electrical resistance] β‘ Ξ© = V/A [Dynamic viscosity] β‘ Pa s [Kinematic viscosity] β‘ m2/s Laminar-Turbulent transition criterion: Forced convection cylindrical pipe flow π π β² 2300 Forced convection along vertical/horizontal plate π π β² 5π105 Forced convection over cylinder/sphere π π β² 2π105 Natural convection along vertical plate π π β² 109
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Correlations for natural Convection Use analogy for mass transfer. Arithmetic mean temperature for properties
Geometry Charact. length
Range of Raleigh No. Nu = f (Ra)
L
RaL < 109
RaL = 104-109
RaL = 1010-1013
entire range
ππ’πΏ = 0.68 +0.670π ππΏ
14
οΏ½1 + οΏ½0.492ππ οΏ½
916οΏ½
49
ππ’πΏ = 0.59 π ππΏ
1/4 ππ’πΏ = 0.1 π ππΏ
1/3
ππ’πΏ =
β
ββββ
0.825 +0.387π ππΏ
16
οΏ½1 + οΏ½0.492ππ οΏ½
916οΏ½
827
β
ββββ
2
L
Use vertical plate equations for the upper surface of the cold plate and the lower surface for the hot plate Replace g by g cos(ΞΈ) for 0 < ΞΈ < 60o
π΄π /π
RaL = 104-107
RaL = 107-1011
RaL = 105-1011
ππ’πΏ = 0.54 π ππΏ1/4
ππ’πΏ = 0.15 π ππΏ
1/3
ππ’πΏ = 0.27 π ππΏ1/4
L
A vertical cylinder can be treated as a vertical plate when
π· β₯35πΏπΊππΏ
1/4
D π ππ· β€ 1012 ππ’π· =
β
ββββ
0.6 +0.387π ππ·
16
οΏ½1 + οΏ½0.559ππ οΏ½
916οΏ½
827
β
ββββ
2
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ππ’π· = πΆπ ππ·π
with
D
π ππ· β₯ 1011 ππ β₯ 0.7
ππ β 1 1 < π ππ· < 105
ππ’π· = 2 +0.589π ππ·
14
οΏ½1 + οΏ½0.469ππ οΏ½
916οΏ½
49
ππ’π· = 2 + 0.43π ππ·
1/4
Correlations for forced convection in internal flow For mass transfer, use appropriate analogy.
Geometry Flow regime Restrictions Nu = f (Re,Pr)
Cylindrical pipe of
diameter D or
Non-cylindrical duct with Dh=4Ac/P
Laminar & fully developed
(Graetz solution for long pipes)
Properties are evaluated at arithmetic mean
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Cylindrical pipe of
diameter D
Laminar within velocity & thermal
entrance length (short pipes)
0.0044 β€ οΏ½ππππ€οΏ½ β€ 9.75
0.6 β€ ππ β€ 5
2β€L/Dβ€20
20<L/D<60
ππ’π· = 1.86 οΏ½ππ π·πΏοΏ½1/3
οΏ½ππππ€οΏ½0.14
ππ=viscosity bulk temperature ππ€=viscosity wall temperature Al other properties are evaluated at bulk temperature
βπΏββ
= 1 + (π· πΏβ )0.7
βπΏββ
= 1 + 6(π· πΏβ )
ββ= value for fully-developed regime
Cylindrical pipe of
diameter D
Turbulent & fully developed
0.7 β€ ππ β€ 100 π π > 104
L/D>60
0.7 β€ ππ β€ 17000 π π > 104
L/D>60
ππ’π· = 0.023π ππ·45πππ
n=0.4 for heating (Ts>Tm) n=0.3 for cooling (Ts<Tm) properties at arithmetic mean
πππ· = 0.023π ππ·β15 ππβ
23 οΏ½ππππ€οΏ½0.14
All properties, except ΞΌw evaluated at bulk temperature
Correlations for forced convection for external flow Plates: For mass transfer, use appropriate analogies. Spheres, Cylinders: Analogies break down, use appropriate correlation
Geometry Flow regime Restrictions Nu = f (Re,Pr)
Flat plate of length L
Laminar (Blasius solution)
ππ β₯ 0.6 or
0.6 β€ ππ β€ 2500 π π < 2 β 105
ππ’π₯ = 0.332Rex12 ππ
13
ππ’πΏ = 0.664ReL1/2 ππ1/3
Properties are evaluated at arithmetic mean
Flat plate of length L Turbulent π π > 3 β 106
ππ’π₯ = 0.0288π ππ₯4/5ππ1/3 ππ’πΏ = 0.036π ππΏ4/5ππ1/3
Properties are evaluated at arithmetic mean
Cylinder of diameter D in crossflow
Laminar Pr = 1
ππ’π· = π΅ π ππ·π ππ1/3
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Cylinder of diameter D in crossflow
Laminar & turbulent Pr > 0.2
ππ’π· = 0.3 +0.62π ππ·
12 ππ
13
οΏ½1 + (0.4/ππ)23 οΏ½
14
οΏ½1 + οΏ½π ππ·
282,000οΏ½58οΏ½
4/5
Properties are evaluated at arithmetic mean
Sphere of diameter D Laminar
20 β² π ππ· β² 105
0.71 β€ ππ β€ 380 3.5 < π ππ· < 7.6 β 104
ππ’π· β 0.31 (π ππ·)0.6
ππ’π· = 2 + ππ0.4 οΏ½πβππ οΏ½1/4
οΏ½0.4π ππ·12 + 0.06π ππ·
23 οΏ½
Properties are evaluated at Tβ, except ΞΌs which is evaluated at Ts
Falling spherical droplet of diameter D
ππ’π· = 2 + 0.6π ππ·
1/2ππ1/3
Sphere of diameter D
For flux of species A from a sphere
into an infinite sink of stagnant fluid B
πβπ· = 2
For mass transfer into liquid streams
πππ΄π΄ < 10,000 πππ΄π΄ > 10,000
πβ = οΏ½4 + 1.21πππ΄π΄23 οΏ½
12
πβ = 1.01 πππ΄π΄1/3
For mass transfer into gas streams
2 < π π < 800 0.6 < ππ < 2.7
or 1500 < π π < 12000
0.6 < ππ < 1.85
πβ = 2 + 0.552π π1/2ππ1/3
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List of dimensionless groups L= characteristic length scale external flow; R = characteristic length scale internal flow/particle u= characteristic velocity; HT = heat transfer; MT = mass transfer, D = diffusivity Dimensionless Groups Definition Interpretation Archimedes number
π΄π =πππΏ3βππ2
=ππΏ3
π2
gravitational force / viscous force
Arrhenius number πΌ =πΈππ π
activation energy / thermal energy
Biot number (heat) π΅π΅ =βπΏπ
convective HT / conductive HT
Biot number (mass) π΅π΅π =βππΏπ
convective MT / diffusive MT
Bodenstein number π΅π΅ =π’ πΏπ·ππ₯
convective MT / axial diffusive MT (Peclet number for chemical reactors, π·ππ₯ =axial diffusion coefficient)
Bond Number π΅π΅ =
ποΏ½ππ β πποΏ½πΏ2
π
gravitational force / capillary force
Brinkmann number π΅π =
ππ’2
π(π π€ β π 0) viscous dissipation / thermal conduction
Capillary number πΆπ =π π’π
viscous force / capillary (surface tension) force
Dean number π·π =
π’ πΏποΏ½πΏπ
= π ποΏ½πΏπ
centrifugal force / viscous force
Eckert number πΈπ =
π’2
ππ(π π€ β π 0) kinetic energy flow / boundary layer enthalpy
Euler number πΈπ’ =Ξπππ’2
pressure force / inertial force
Fourier number HT πΉπ΅ =
πΌππΏ2
=π π
π ππ πΏ2
heat conduction / enthalpy change; also dimensionless time
Fourier number MT πΉπ΅π =π· ππΏ2
diffusion rate / species accumulation; dimensionless time
Inertial friction factor πππ =ΞππΏ
π ππ’2
specific pressure drop / inertial force
Viscous friction factor ππ£ππ =
ΞππΏπ 2
π π’
specific pressure drop / viscous force
Froude number πΉπ =
π’2
π πΏ
inertial force / gravitational force
Galileo number πΊπ =
πππΏ3 π2
= π ππΏ2ππ π’
Reynolds x gravity force / viscous force
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Graetz number HT πΊπ§ =
π 2 π π’ ππ πΏ π
=π πΏπ π ππ
thermal capacity flow / conductive HT
Graetz number MT πΊπ§π =
π 2 π’ πΏ π·
=π πΏπ π ππ
mass capacity (flow) / diffusive MT
Grashof number πΊπ =
π π½ (π π€ β π 0)πΏ3 π2
buoyant force / viscous force
Knudsen number πΎπΎ =π πΏ
length of free mean path / characteristic length
Lewis number πΏπ =πΌ π·
thermal diffusivity / mass diffusivity
Mach number ππ =π’
π’π π πππ velocity / speed of sound
Nusselt number ππ’ =β πΏπ
convective HT / conductive HT (at boundaries)
Ohnesorge number πβ =
ποΏ½ππΏ π
=βπππ π
viscous force / SQRT(inertial force x capillary force)
Peclet number HT ππ =π£ πΏπΌ
=π’ π ππ πΏ
π= π π ππ
convective HT / diffusive HT (in bulk liquid)
Peclet number MT πππ =π’ πΏπ·
= π π ππ convective MT / diffusive MT in bulk liquid
Prandtl number ππ =ππΌ
=π πππ
viscous diffusivity / thermal diffusivity
Raleigh number π π = πΊπ ππ natural convection HT / conductive HT
Reynolds number π π =π’ πΏπ
=π’ πΏπ/π
inertial force / viscous force
Schmidt number ππ =ππ·
=π/ππ·
momentum diffusivity / mass diffusivity
Sherwood number πβ =ππ πΏ π·
convective MT / diffusive MT (at boundaries)
Stanton number HT ππ =ππ’ π π ππ
=ππ’ππ
convective HT / heat capacity (at boundaries)
Stanton number MT πππ =πβ π π ππ
=πβπππ
convective MT / mass capacity (at boundaries)
Stokes number πππ =πππ’π
particle relaxation time / convective time scale
Strouhal number ππ =π πΏ π’
characteristic frequency / characteristic timescale-1
Weber number ππ =
ππ’2ππΏ
inertial force / capillary force
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