Heat and Mass Transfer CHEE330 – Formula Sheet

Version 05/12/2014 Page 8

Fourier’s law

𝑞"��� =𝑞𝐴

= −𝑘𝑘𝑘 𝑞 Heat transfer rate through area A [W] 𝑞"��� = 𝑞

𝐴 Heat flux or heat transfer rate per unit area perpendicular to the

transport direction [W/m2=J/(s m2] 𝐴 Area perpendicular to heat flux [m2] k Thermal conductivity [W/(m K)] ∇𝑘 Temperature gradient (driving force) [K/m] 1-Dimensional Fourier’s law for different coordinate systems Fourier’s 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

Fourier’s law 𝑞"𝑥 = −𝑘𝑑𝑘𝑑𝑑

𝑞"𝑟 = −𝑘𝑑𝑘𝑑𝑑

𝑞"𝑟 = −𝑘𝑑𝑘𝑑𝑑

Heat flux 𝒒" 𝑘∆𝑘𝐿

𝑘∆𝑘

𝑑𝑟𝑟 �𝑑2𝑑1�

𝑘∆𝑘

𝑑2 �1𝑑1− 1𝑑2�

Heat transfer rate 𝒒 𝑘𝐴

∆𝑘𝐿

2𝜋𝐿𝑘∆𝑘

𝑟𝑟 �𝑑2𝑑1�

4𝜋𝑘∆𝑘

�1𝑑1− 1𝑑2�

Thermal resistance

Rcond #

𝐿𝑘𝐴

𝑟𝑟 �𝑑2𝑑1

�

2𝜋𝐿𝑘

�1𝑑1− 1𝑑2�

4𝜋𝑘

#Ar=2πrL for cylindrical, Ar=4πr2 for spherical coordinates, r1=rin, r2=rout Radiation Stefan-Boltzmann law for an ideal radiator (black body)

𝑞𝑒𝑒" = 𝐸 = 𝜎𝑘𝑠4 𝑞𝑒𝑒" = radiation/heat flux emitted from the surface Ts = absolute temperature of the surface [K] σ = Stefan-Boltzmann constant

For a real (non-ideal) surface 𝑞𝑒𝑒" = 𝐸 = 𝜀𝜎𝑘𝑠4

ε= emissivity [-] -> black bodies: 𝜀=1, real surface: 0<𝜀<1 Irradiation

𝑞𝑖𝑖𝑖" = 𝐺 = 𝛼 𝜎 𝑘𝑠𝑠𝑟4 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]

α = absorptivity of the surface [0< α <1], for a “grey” surface α=ε

Net radiation exchange 𝑞𝑟𝑟𝑟" = 𝐸 − 𝐺 = 𝜀𝜎𝑘𝑠4 − 𝛼 𝜎 𝑘𝑠𝑠𝑟4 = ℎ𝑟(𝑘𝑠 − 𝑘𝑠𝑠𝑟)

Radiative heat transfer coefficient for grey surface ℎ𝑟 = 𝜎𝜀(𝑘𝑠 + 𝑘𝑠𝑠𝑟)(𝑘𝑠2 − 𝑘𝑠𝑠𝑟2 ) [W/m2K]

Thermal circuits

𝑞 =𝑂𝑂𝑂𝑑𝑂𝑟𝑟 𝑑𝑑𝑑𝑂𝑑𝑟𝑑 𝑓𝑓𝑑𝑓𝑂

𝑅𝑂𝑅𝑑𝑅𝑅𝑂𝑟𝑓𝑂=∆𝑘𝑅

R = thermal resistance [K/W] Conductive resistance

Rcond = depends on geometry, see table left Convective resistance

𝑅𝑖𝑐𝑖𝑐 =1ℎ 𝐴

Radiative resistance

𝑅𝑟𝑟𝑟 =1

ℎ𝑟 𝐴

Thermal contact resistance

𝑅"𝑡,𝑖 =𝑘𝐴 − 𝑘𝐵𝑞𝑥"

TA,B = temperature contact surface A,B [K]

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Resistance in series (q=const):

𝑅𝑡𝑐𝑡 = 𝑅1 + 𝑅2+. . +𝑅𝑖 = �𝑅𝑖𝑖

Resistances in parallel (ΔT =const): 1𝑅𝑡𝑐𝑡

=1𝑅1

+1𝑅2

+. . +1𝑅𝑖

= �1/𝑅𝑖𝑖

Ideal gas law

𝑝 𝑉 = 𝑟 𝑅 𝑘 =𝑚𝑀𝑅 𝑘

𝑝 = 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 ≤ 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 𝛱1 = 𝐹(𝛱2,𝛱3. .𝛱𝑘)

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 (Fick’s Law)

𝑓 𝑓𝑑 𝜌 = 𝑓𝑓𝑟𝑅𝑅 𝐽𝐴��� = −𝐷𝐴𝐵𝑘𝑓𝐴 𝐽𝐴,𝑧 = −𝐷𝐴𝐵𝑑𝑓𝐴𝑑𝑑

Mass flux (Fick’s Law) 𝑓 𝑓𝑑 𝜌 = 𝑓𝑓𝑟𝑅𝑅 𝚥𝐴��� = −𝐷𝐴𝐵𝑘𝜌𝐴 𝑗𝐴,𝑧 = −𝐷𝐴𝐵

𝑑𝜌𝐴𝑑𝑑

Molar flux (de Groot) 𝐽𝐴��� = −𝑓𝐷𝐴𝐵𝑘𝑦𝐴 𝐽𝐴,𝑧 = −𝑓𝐷𝐴𝐵𝑑𝑦𝐴𝑑𝑑

Mass flux (de Groot) 𝚥𝐴��� = −𝜌𝐷𝐴𝐵𝑘𝜔𝐴 𝑗𝐴,𝑧 = −𝜌𝐷𝐴𝐵𝑑𝜔𝐴𝑑𝑑

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 𝑁𝐴���� = −𝐷𝐴𝐵𝑓𝑘𝑦𝐴 +𝑦𝐴(𝑁𝐴���� + 𝑁𝐵����� )

𝑁𝐴,𝑧 = −𝐷𝐴𝐵𝑓𝑑𝑦𝐴𝑑𝑑

+𝑦𝐴(𝑁𝐴,𝑧 + 𝑁𝐵,𝑧)

Mass flux 𝑟𝐴���� = −𝜌𝐷𝐴𝐵𝑘𝜔𝐴 +𝜔𝐴(𝑟𝐴���� + 𝑟𝐵���� )

𝑟𝐴,𝑧 = −𝜌𝐷𝐴𝐵𝑑𝜔𝐴𝑑𝑑

+𝜔𝐴(𝑟𝐴,𝑧 + 𝑟𝐵,𝑧) Molar flux for

equimolar counter diffusion

(𝑁𝐴���� = −𝑁𝐵����� )

𝑁𝐴���� = − DAB𝑓𝑘𝑦𝐴 𝑁𝐴,𝑧 = DAB𝑓�yA,1 − yA,2�

𝑑2 − 𝑑1

Molar flux for unimolecular

diffusion stagnant film (𝑁𝐵����� = 0)

𝑁𝐴���� = −𝐷𝐴𝐵𝑓

1 − 𝑦𝐴𝑘𝑦𝐴

𝑁𝐴,𝑧 = 𝐷𝐴𝐵𝑓

(𝑑2 − 𝑑1) ∙

ln�1 − 𝑦𝐴,2

1 − 𝑦𝐴,1�

Control volume balance on rate basis In a defined control volume, there is

ACCUMULATION = INPUT - OUTPUT + GENERATION Energy:

𝑟𝐸𝑠𝑟𝑡

= ��𝑖𝑖- ��𝑐𝑠𝑡 + ��𝑔 𝐸𝑠= stored energy [J] ��𝑖𝑖= ingoing energy rate [W] ��𝑐𝑠𝑡= outgoing energy rate [W] ��𝑔= generated energy rate [W]

Mass Species A 𝑑𝑀𝐴

𝑑𝑅= ��𝐴,𝑖𝑖 − ��𝐴,𝑐𝑠𝑡 + ��𝐴,𝑔

𝑀𝐴 = stored mass of A [kg] ��𝐴,𝑖𝑖 = ingoing mass rate of A [kg/s] ��𝐴,𝑐𝑠𝑡 = outgoing mass rate of A [kg/s] ��𝐴,𝑔 = generated mass rate of A [kg/s]

Continious flow system

���𝑖𝑒𝑡 + ��𝑖𝑒𝑡� = �� �(ℎ2 − ℎ1) +12

(𝑂22 − 𝑂12) + 𝑑(𝑑2 − 𝑑1)� ��𝑖𝑒𝑡 = net heat rate added to CV [W] ��𝑖𝑒𝑡= net rate of work done in CV [W] �� = mass flow rate [kg/s] 𝑑𝑖 = height [m] 𝑂𝑖= velocity [m/s] ℎ𝑖 = 𝑓𝑝 𝑘𝑖 = specific enthalpy [J/(kg K)] 1 = inlet, 2=outlet

Differential Equations of Heat Transfer for k = 𝒇(𝒙�� )

𝜌𝑓𝑝𝜕𝑘𝜕𝑅

= 𝑘 ∙ (𝑘𝑘𝑘 ) + ��𝑔

for k = constant

𝜕𝑘𝜕𝑅

= 𝛼∆𝑘 +��𝑔𝜌𝑓𝑝

��𝑔= volumetric generation term [W/m3] 𝜌 = density [kg/m3] α = k

ρ cp = thermal diffusivity [m2/s] 𝑓𝑝= specific heat capacity [kJ/(kg K)]

Boundary condition of first kind - Dirichlet condition Constant Temperature

𝑘(𝑑 = 𝑑0, 𝑅) = 𝑓𝑓𝑟𝑅𝑅. Boundary condition of second kind - Neumann condition Constant gradient at a boundary (=constant flux)

𝑑𝑘𝑑𝑑�𝑥=𝑥0

= 𝑓𝑓𝑟𝑅𝑅.

Boundary condition of third kind - Robin boundary condition The gradient at a boundary is described with a function (e.g. Newton’s Law of cooling)

𝑑𝑘𝑑𝑑�𝑥=𝑥0

= 𝑓(𝑘)

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Differential Equations of Mass Transfer 𝜕𝑓𝐴𝜕𝑅

= −𝑘 ∙ 𝑁��𝐴 + 𝑅𝐴

𝑁��𝐴 can be either the purely diffusive flux 𝐽𝐴��� or absolute flux 𝑁��𝐴 of A 𝑅𝐴 = volumetric rate of mass generation [mol/(s m3)]

𝜕𝑓𝐴𝜕𝑅

= 𝑘 ∙ (𝐷𝐴𝐵𝑓𝑘𝑦𝐴) − 𝑘 ∙ �𝑓𝐴𝑉� � + 𝑅𝐴

𝑉� =molar-average velocity [m/s] Boundary condition of first kind - Dirichlet condition Constant Temperature

𝑓𝐴(𝑑 = 𝑑0, 𝑅) = 𝑓𝑓𝑟𝑅𝑅. Boundary condition of second kind - Neumann condition Constant gradient at a boundary (=constant flux)

𝑑𝑓𝐴𝑑𝑑

�𝑥=𝑥0

= 𝑓𝑓𝑟𝑅𝑅.

Boundary condition of third kind - Robin boundary condition The gradient at a boundary is described with a function

𝑑𝑓𝐴𝑑𝑑

�𝑥=𝑥0

= 𝑓(𝑓𝐴)

Other Boundary conditions for mass transfer Evaporation and sublimation (Raoult’s Law)

𝑝𝐴,𝑠 = 𝑑𝐴 𝑃𝐴,𝑠𝑟𝑡 𝑝𝐴,𝑠 = 𝑦𝐴,𝑠 𝑃= partial pressure of A in gas at the surface [bar]

𝑃𝐴,𝑠𝑟𝑡 = saturation (vapor) pressure at the surface Solubility of gases in liquids (Henry’s Law)

𝑝𝐴 = 𝐻𝑑𝐴 𝐻= Henry constant [Pa] Solubility of gases in solids

𝑓𝐴,𝑠𝑐𝑠𝑖𝑟 = 𝑆 𝑝𝐴 𝑆= 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 (𝒓,𝜽, 𝒛)

Spherical (𝒓,𝜽,𝝓)

Gradient 𝜵𝒇

⎝

⎜⎜⎜⎛

𝜕𝑓𝜕𝑑𝜕𝑓𝜕𝑦𝜕𝑓𝜕𝑑⎠

⎟⎟⎟⎞

⎝

⎜⎜⎛

𝜕𝑓𝜕𝑑

1𝑑𝜕𝑓𝜕θ𝜕𝑓𝜕𝑑 ⎠

⎟⎟⎞

⎝

⎜⎜⎜⎛

𝜕𝑓𝜕𝑑

1𝑑𝜕𝑓𝜕𝜕

1𝑑 𝑅𝑑𝑟 (𝜕)

𝜕𝑓𝜕𝜕⎠

⎟⎟⎟⎞

Laplace 𝜵𝟐𝒇 = 𝚫𝒇

�𝜕2𝑓𝜕𝑑2

+𝜕2𝑓𝜕𝑦2

+𝜕2𝑓𝜕𝑑2

�

�1𝑑

𝜕𝜕𝑑�𝑑𝜕𝑓𝜕𝑑�

+1𝑑2𝜕2𝑓𝜕𝜕2

+𝜕2𝑓𝜕𝑑2

�

�1𝑑2

𝜕𝜕𝑑�𝑑2

𝜕𝑓𝜕𝑑�

+1

𝑑2𝑅𝑑𝑟 (𝜕)𝜕𝜕𝜕

�𝑅𝑑𝑟 (𝜕)𝜕𝑓𝜕𝜕�

+1

𝑑2 𝑅𝑑𝑟2(𝜕) 𝜕2𝑓𝜕𝜕2�

Divergence 𝜵 ∙ 𝑭��

�𝜕𝐹𝑥𝜕𝑑

+𝜕𝐹𝑦𝜕𝑑

+𝜕𝐹𝑧𝜕𝑑

�

�1𝑑𝜕(𝑑 𝐹𝑟)𝜕𝑑

+1𝑑𝜕𝐹θ𝜕𝜕

+𝜕𝐹𝑧𝜕𝑑

�

�1𝑑2𝜕(𝑑2𝐹𝑟)𝜕𝑑

+1

𝑑 𝑅𝑑𝑟 (𝜕)𝜕(𝐹𝜃 𝑅𝑑𝑟 (𝜕))

𝜕𝜕

+1

𝑑 𝑅𝑑𝑟 (𝜕) 𝜕𝐹𝜙𝜕𝜕

�

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Convective heat transfer Newton’s law of Cooling

𝑞𝐴

= 𝑞" = ℎ∆𝑘 h = convective HT coefficient [W/(m2K)] ∆𝑘 = temperature difference [K] Internal Flow

𝑞𝑖𝑐𝑖𝑐 = ��𝑓𝑝�𝑘𝑒,𝑐 − 𝑘𝑒,𝑖� = ℎ 𝐴 ∆𝑘𝑠𝑒 Logarithmic temperature difference

∆𝑘𝑠𝑒 =∆𝑘𝑐 − ∆𝑘𝑖

𝑟𝑟 �∆𝑘𝑐∆𝑘𝑖�

Constant surface temperature ∆𝑘𝑐 = 𝑘𝑠 − 𝑘𝑐𝑠𝑡 ∆𝑘𝑖 = 𝑘𝑠 − 𝑘𝑖𝑖

Energy balance results in

𝑟𝑟 �𝑘𝑒,𝑐𝑠𝑡 − 𝑘𝑠𝑘𝑒,𝑖𝑖 − 𝑘𝑠

� +ℎ

𝑂𝑟𝑐𝑔𝜌𝑓𝑝4𝐿𝐷

= 0

Constant external temperature use modified Newton’s Law

q =∆𝑘𝑠𝑒𝑅𝑡𝑐𝑡

𝑅𝑡𝑐𝑡 = total resistance of convective and conductive HT ∆𝑘𝑠𝑒 built with

∆𝑘𝑐 = 𝑘∞ − 𝑘𝑐𝑠𝑡 ∆𝑘𝑖 = 𝑘∞ − 𝑘𝑖𝑖

Constant heat flux: Local mean temperature of the fluid:

𝑘𝑒(𝑑) = 𝑘𝑒,𝑖 +𝑞𝑠"𝑃��𝑓𝑃

𝑑

Tm,i= mean temperature inlet [K] P=cross section perimeter [m]

m= mass flow rate [kg/s] cp = specific heat capacity [kJ/(kg K)] qs“= heat flux at the surface [W/m2] Average heat coefficient

ℎ𝐿 =1𝐿� ℎ𝑥𝑑𝑑𝐿

0

ℎ𝐿= average heat transfer coefficient over a spatial dimension L ℎ𝑥= local heat transfer coefficient at a certain position x Convective mass transfer

𝑁𝐴 = 𝑘𝑖∆𝑓𝐴 NA = molar convective mass transfer flux [mol/(m2s] 𝑘𝑖= concective mass transfer coefficient [m/s] ∆𝑓𝐴= 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

𝐶𝑓,𝑥 =2𝜏𝑆,𝑥

𝜌𝑂∞2

𝜏𝑆,𝑥 = Local shear stress at position x [N/m2] Use average skin friction for analogy of average coefficients

𝐶𝑓,𝐿 =2𝜏𝑆,𝐿

𝜌𝑂∞2

𝜏𝑆,𝐿 = 𝐹𝐴

= Average shear stress = Drag force per surface area over spatial dimension L [N/m2] Reynolds analogy

𝑆𝑅 =ℎ

𝜌𝑂∞𝑓𝑝=𝐶𝑓2

= 𝑆𝑅𝑒 =𝑘𝑖𝑂∞

valid for Blasius solution (laminar flow) of the horizontal plate and Pr=1 and Sc=1

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local skin friction

𝐶𝑓,𝑥 =0.664

�𝑅𝑂𝑥

average skin friction for averaged coefficients

𝐶𝑓,𝐿 =1.328

�𝑅𝑂𝐿

Chilton-Colburn analogy For laminar and turbulent flow where is no form drag such as flow over flat plate and internal flows

𝑗𝐻 = 𝑗𝐷 =𝐶𝑓2

𝑗𝐻 = 𝑆𝑅 𝑃𝑑2/3

valid for 0.5<Pr<50

𝑗𝐷 =𝑘𝑖𝑂∞

𝑆𝑓2/3

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

𝑆𝑅 =𝐶𝑓/2

1 + 5�𝐶𝑓 2⁄ (𝑃𝑑 − 1)

for mass transfer Stanton number use Sc instead of Pr. Constants g = Gravitational acceleration =9.81 m2/s kB= Boltzmann constant =1.38 × 10-23J/K R = Universal gas constant = 8.3143 J/(mol K)

σ = Stefan-Boltzmann constant σ = 5.67x10-8 W/(m2K4) NA = Avogadro number 6.022 × 1023 mol−1

Units of selected physical quantities: [Pressure] ≡ atm (standard) = 101325 Pa bar = 105 Pa Pa = N/m2 [Force] ≡ N = kg m/s2 [Work] ≡ J = N m [Power] ≡ W = J/s [Charge] ≡ C [Current] ≡ A = C/s [Voltage] ≡ V = J/C [Electrical resistance] ≡ Ω = V/A [Dynamic viscosity] ≡ Pa s [Kinematic viscosity] ≡ m2/s Laminar-Turbulent transition criterion: Forced convection cylindrical pipe flow 𝑅𝑂 ≲ 2300 Forced convection along vertical/horizontal plate 𝑅𝑂 ≲ 5𝑑105 Forced convection over cylinder/sphere 𝑅𝑂 ≲ 2𝑑105 Natural convection along vertical plate 𝑅𝑂 ≲ 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

𝑁𝑢𝐿 = 0.68 +0.670𝑅𝑎𝐿

14

�1 + �0.492𝑃𝑃 �

916�

49

𝑁𝑢𝐿 = 0.59 𝑅𝑎𝐿

1/4 𝑁𝑢𝐿 = 0.1 𝑅𝑎𝐿

1/3

𝑁𝑢𝐿 =

⎝

⎜⎜⎜⎛

0.825 +0.387𝑅𝑎𝐿

16

�1 + �0.492𝑃𝑃 �

916�

827

⎠

⎟⎟⎟⎞

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(θ) for 0 < θ < 60o

𝐴𝑠/𝑃

RaL = 104-107

RaL = 107-1011

RaL = 105-1011

𝑁𝑢𝐿 = 0.54 𝑅𝑎𝐿1/4

𝑁𝑢𝐿 = 0.15 𝑅𝑎𝐿

1/3

𝑁𝑢𝐿 = 0.27 𝑅𝑎𝐿1/4

L

A vertical cylinder can be treated as a vertical plate when

𝐷 ≥35𝐿𝐺𝑃𝐿

1/4

D 𝑅𝑎𝐷 ≤ 1012 𝑁𝑢𝐷 =

⎝

⎜⎜⎜⎛

0.6 +0.387𝑅𝑎𝐷

16

�1 + �0.559𝑃𝑃 �

916�

827

⎠

⎟⎟⎟⎞

2

Heat and Mass Transfer CHEE330 – Formula Sheet

Version 5/12/2014 Page 15

𝑁𝑢𝐷 = 𝐶𝑅𝑎𝐷𝑛

with

D

𝑅𝑎𝐷 ≥ 1011 𝑃𝑃 ≥ 0.7

𝑃𝑃 ≈ 1 1 < 𝑅𝑎𝐷 < 105

𝑁𝑢𝐷 = 2 +0.589𝑅𝑎𝐷

14

�1 + �0.469𝑃𝑃 �

916�

49

𝑁𝑢𝐷 = 2 + 0.43𝑅𝑎𝐷

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

Heat and Mass Transfer CHEE330 – Formula Sheet

Version 5/12/2014 Page 16

Cylindrical pipe of

diameter D

Laminar within velocity & thermal

entrance length (short pipes)

0.0044 ≤ �𝜇𝑏𝜇𝑤� ≤ 9.75

0.6 ≤ 𝑃𝑃 ≤ 5

2≤L/D≤20

20<L/D<60

𝑁𝑢𝐷 = 1.86 �𝑃𝑃 𝐷𝐿�1/3

�𝜇𝑏𝜇𝑤�0.14

𝜇𝑏=viscosity bulk temperature 𝜇𝑤=viscosity wall temperature Al other properties are evaluated at bulk temperature

ℎ𝐿ℎ∞

= 1 + (𝐷 𝐿⁄ )0.7

ℎ𝐿ℎ∞

= 1 + 6(𝐷 𝐿⁄ )

ℎ∞= value for fully-developed regime

Cylindrical pipe of

diameter D

Turbulent & fully developed

0.7 ≤ 𝑃𝑃 ≤ 100 𝑅𝑃 > 104

L/D>60

0.7 ≤ 𝑃𝑃 ≤ 17000 𝑅𝑃 > 104

L/D>60

𝑁𝑢𝐷 = 0.023𝑅𝑃𝐷45𝑃𝑃𝑛

n=0.4 for heating (Ts>Tm) n=0.3 for cooling (Ts<Tm) properties at arithmetic mean

𝑆𝑆𝐷 = 0.023𝑅𝑃𝐷−15 𝑃𝑃−

23 �𝜇𝑏𝜇𝑤�0.14

All properties, except μw 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)

𝑃𝑃 ≥ 0.6 or

0.6 ≤ 𝑆𝑆 ≤ 2500 𝑅𝑃 < 2 ⋅ 105

𝑁𝑢𝑥 = 0.332Rex12 𝑃𝑃

13

𝑁𝑢𝐿 = 0.664ReL1/2 𝑃𝑃1/3

Properties are evaluated at arithmetic mean

Flat plate of length L Turbulent 𝑅𝑃 > 3 ⋅ 106

𝑁𝑢𝑥 = 0.0288𝑅𝑃𝑥4/5𝑃𝑃1/3 𝑁𝑢𝐿 = 0.036𝑅𝑃𝐿4/5𝑃𝑃1/3

Properties are evaluated at arithmetic mean

Cylinder of diameter D in crossflow

Laminar Pr = 1

𝑁𝑢𝐷 = 𝐵 𝑅𝑃𝐷𝑛 𝑃𝑃1/3

Heat and Mass Transfer CHEE330 – Formula Sheet

Version 5/12/2014 Page 17

Cylinder of diameter D in crossflow

Laminar & turbulent Pr > 0.2

𝑁𝑢𝐷 = 0.3 +0.62𝑅𝑃𝐷

12 𝑃𝑃

13

�1 + (0.4/𝑃𝑃)23 �

14

�1 + �𝑅𝑃𝐷

282,000�58�

4/5

Properties are evaluated at arithmetic mean

Sphere of diameter D Laminar

20 ≲ 𝑅𝑃𝐷 ≲ 105

0.71 ≤ 𝑃𝑃 ≤ 380 3.5 < 𝑅𝑃𝐷 < 7.6 ⋅ 104

𝑁𝑢𝐷 ≈ 0.31 (𝑅𝑃𝐷)0.6

𝑁𝑢𝐷 = 2 + 𝑃𝑃0.4 �𝜇∞𝜇𝑠�1/4

�0.4𝑅𝑃𝐷12 + 0.06𝑅𝑃𝐷

23 �

Properties are evaluated at T∞, except μs which is evaluated at Ts

Falling spherical droplet of diameter D

𝑁𝑢𝐷 = 2 + 0.6𝑅𝑃𝐷

1/2𝑃𝑃1/3

Sphere of diameter D

For flux of species A from a sphere

into an infinite sink of stagnant fluid B

𝑆ℎ𝐷 = 2

For mass transfer into liquid streams

𝑃𝑃𝐴𝐴 < 10,000 𝑃𝑃𝐴𝐴 > 10,000

𝑆ℎ = �4 + 1.21𝑃𝑃𝐴𝐴23 �

12

𝑆ℎ = 1.01 𝑃𝑃𝐴𝐴1/3

For mass transfer into gas streams

2 < 𝑅𝑃 < 800 0.6 < 𝑆𝑆 < 2.7

or 1500 < 𝑅𝑃 < 12000

0.6 < 𝑆𝑆 < 1.85

𝑆ℎ = 2 + 0.552𝑅𝑃1/2𝑆𝑆1/3

Heat and Mass Transfer CHEE330 – Formula Sheet

Version 5/12/2014 Page 18

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

𝐴𝑃 =𝑔𝑔𝐿3∆𝑔𝜇2

=𝑔𝐿3

𝜈2

gravitational force / viscous force

Arrhenius number 𝛼 =𝐸𝑎𝑅𝑅

activation energy / thermal energy

Biot number (heat) 𝐵𝐵 =ℎ𝐿𝑘

convective HT / conductive HT

Biot number (mass) 𝐵𝐵𝑚 =ℎ𝑚𝐿𝑘

convective MT / diffusive MT

Bodenstein number 𝐵𝐵 =𝑢 𝐿𝐷𝑎𝑥

convective MT / axial diffusive MT (Peclet number for chemical reactors, 𝐷𝑎𝑥 =axial diffusion coefficient)

Bond Number 𝐵𝐵 =

𝑔�𝑔𝑙 − 𝑔𝑔�𝐿2

𝜎

gravitational force / capillary force

Brinkmann number 𝐵𝑃 =

𝜇𝑢2

𝑘(𝑅𝑤 − 𝑅0) viscous dissipation / thermal conduction

Capillary number 𝐶𝑎 =𝜇 𝑢𝜎

viscous force / capillary (surface tension) force

Dean number 𝐷𝑃 =

𝑢 𝐿𝜈�𝐿𝑅

= 𝑅𝑃�𝐿𝑅

centrifugal force / viscous force

Eckert number 𝐸𝑆 =

𝑢2

𝑆𝑝(𝑅𝑤 − 𝑅0) kinetic energy flow / boundary layer enthalpy

Euler number 𝐸𝑢 =Δ𝑝𝑔𝑢2

pressure force / inertial force

Fourier number HT 𝐹𝐵 =

𝛼𝑆𝐿2

=𝑘 𝑆

𝑔 𝑆𝑝 𝐿2

heat conduction / enthalpy change; also dimensionless time

Fourier number MT 𝐹𝐵𝑚 =𝐷 𝑆𝐿2

diffusion rate / species accumulation; dimensionless time

Inertial friction factor 𝑓𝑖𝑛 =Δ𝑝𝐿

𝑅𝑔𝑢2

specific pressure drop / inertial force

Viscous friction factor 𝑓𝑣𝑖𝑠 =

Δ𝑝𝐿𝑅2

𝜇 𝑢

specific pressure drop / viscous force

Froude number 𝐹𝑃 =

𝑢2

𝑔 𝐿

inertial force / gravitational force

Galileo number 𝐺𝑎 =

𝑔𝑔𝐿3 𝜇2

= 𝑅𝑃𝐿2𝑔𝜇 𝑢

Reynolds x gravity force / viscous force

Heat and Mass Transfer CHEE330 – Formula Sheet

Version 5/12/2014 Page 19

Graetz number HT 𝐺𝑧 =

𝑅2 𝑔 𝑢 𝑆𝑝 𝐿 𝑘

=𝑅𝐿𝑅𝑃 𝑃𝑃

thermal capacity flow / conductive HT

Graetz number MT 𝐺𝑧𝑚 =

𝑅2 𝑢 𝐿 𝐷

=𝑅𝐿𝑅𝑃 𝑆𝑆

mass capacity (flow) / diffusive MT

Grashof number 𝐺𝑃 =

𝑔 𝛽 (𝑅𝑤 − 𝑅0)𝐿3 𝜈2

buoyant force / viscous force

Knudsen number 𝐾𝐾 =𝜆 𝐿

length of free mean path / characteristic length

Lewis number 𝐿𝑃 =𝛼 𝐷

thermal diffusivity / mass diffusivity

Mach number 𝑀𝑎 =𝑢

𝑢𝑠𝑠𝑛𝑖𝑠 velocity / speed of sound

Nusselt number 𝑁𝑢 =ℎ 𝐿𝑘

convective HT / conductive HT (at boundaries)

Ohnesorge number 𝑂ℎ =

𝜇�𝑔𝐿 𝜎

=√𝑊𝑃𝑅𝑃

viscous force / SQRT(inertial force x capillary force)

Peclet number HT 𝑃𝑃 =𝑣 𝐿𝛼

=𝑢 𝑔 𝑆𝑝 𝐿

𝑘= 𝑅𝑃 𝑃𝑃

convective HT / diffusive HT (in bulk liquid)

Peclet number MT 𝑃𝑃𝑚 =𝑢 𝐿𝐷

= 𝑅𝑃 𝑆𝑆 convective MT / diffusive MT in bulk liquid

Prandtl number 𝑃𝑃 =𝜈𝛼

=𝜇 𝑆𝑝𝑘

viscous diffusivity / thermal diffusivity

Raleigh number 𝑅𝑎 = 𝐺𝑃 𝑃𝑃 natural convection HT / conductive HT

Reynolds number 𝑅𝑃 =𝑢 𝐿𝜈

=𝑢 𝐿𝜇/𝑔

inertial force / viscous force

Schmidt number 𝑆𝑆 =𝜈𝐷

=𝜇/𝑔𝐷

momentum diffusivity / mass diffusivity

Sherwood number 𝑆ℎ =𝑘𝑠𝐿 𝐷

convective MT / diffusive MT (at boundaries)

Stanton number HT 𝑆𝑆 =𝑁𝑢 𝑅𝑃 𝑃𝑃

=𝑁𝑢𝑃𝑃

convective HT / heat capacity (at boundaries)

Stanton number MT 𝑆𝑆𝑚 =𝑆ℎ 𝑅𝑃 𝑆𝑆

=𝑆ℎ𝑃𝑃𝑚

convective MT / mass capacity (at boundaries)

Stokes number 𝑆𝑆𝑘 =𝑆𝑝𝑢𝑅

particle relaxation time / convective time scale

Strouhal number 𝑆𝑃 =𝑓 𝐿 𝑢

characteristic frequency / characteristic timescale-1

Weber number 𝑊𝑃 =

𝑔𝑢2𝜎𝐿

inertial force / capillary force