SME
Transport Phenomena · Problem type
Prandtl and Schmidt numbers
Use this for “Engine oil at 80 °C has cₚ = 2131 J/(kg·K), μ = 0.032 Pa·s and k = 0.138 W/(m·K). Find Pr and say what it compares.”
Chart
A log scale: each grid line up is 10 times the one below.
This is an example. Type your own values to start a new problem.
Pr
=
cₚ (J/(kg·K))
×
μ (Pa·s)
k (W/(m·K))
Sc
=
μ (Pa·s)
ρ (kg/m³)
×
DAB (mm²/s)
Pr needs cₚ, μ and k; Sc needs μ, the density ρ and the mass diffusivity DAB. The bars compare ν = μ ÷ ρ, α = k ÷ (ρcₚ) and DAB.
ν
Momentum diffusivity
worked out (not typed) · mm²/s
α
Thermal diffusivity
worked out (not typed) · mm²/s
Formulas
Pr
=
cₚ
×
μ
k
6.993
=
4,182
×
0.001
0.598
Sc
=
μ
ρ
× DAB
501
=
0.001
998 × (2 ×
10
-9
)
ν
=
μ
ρ
1.002 ×
10
-6
=
0.001
998
α
=
k
ρ
× cₚ
1.4328 ×
10
-7
=
0.598
998 × 4,182
Formulas are worked in m²/s. The step-by-step shows the conversions.
Assumptions
•
Pr = ν ÷ α compares momentum with heat diffusion; Sc = ν ÷ DAB compares it with mass diffusion.
•
Above 1, momentum spreads faster, so the velocity layer is the thicker one.
•
Read the properties from a table at the fluid’s temperature.
Step-by-step
WE KNOW
cₚ = 4,182 J/(kg·K), μ = 0.001 Pa·s, k = 0.598 W/(m·K), ρ = 998 kg/m³, DAB = 0.002 mm²/s
FIND
Prandtl number (Pr), Schmidt number (Sc), momentum diffusivity (ν), thermal diffusivity (α)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m²/s).
DAB = 0.002 mm²/s = 2 × 10⁻⁹ m²/s (1 m²/s = 10⁶ mm²/s)
Step 2 · Find Prandtl number (Pr)
Use Pr = cₚ × μ ÷ k
Pr
=
ν
α
,
with
ν
=
μ
ρ
and
α
=
k
ρcₚ
;
the
density
cancels.
Pr
=
4,182
×
0.001
0.598
Pr
=
4.182
0.598
Pr = 6.993
Step 3 · Find Schmidt number (Sc)
Use Sc = μ ÷ (ρ × DAB)
Sc
=
ν
DAB
:
the
momentum
diffusivity
μ
ρ
over
the
mass
diffusivity.
Sc
=
0.001
998 × (2 ×
10
-9
)
Sc
=
0.001
1.996 ×
10
-6
Sc = 501
Step 4 · Find momentum diffusivity (ν)
Use ν = μ ÷ ρ
Kinematic viscosity: how fast momentum spreads through the fluid.
ν
=
0.001
998
ν
=
1.002 ×
10
-6
m
2
/s
Step 5 · Find thermal diffusivity (α)
Use α = k ÷ (ρ × cₚ)
Conduction over the heat stored per volume: how fast heat spreads.
α
=
0.598
998 × 4,182
α
=
0.598
4,173,636
α
=
1.4328 ×
10
-7
m
2
/s
Step 6 · Convert the answers
Change each answer back to the unit you chose.
ν = 1.002 × 10⁻⁶ m²/s = 1.002 mm²/s (1 m²/s = 10⁶ mm²/s)
α = 1.4328 × 10⁻⁷ m²/s = 0.1433 mm²/s (1 m²/s = 10⁶ mm²/s)
Check
Put the numbers back in. Both sides should match.
6.993
=
4,182
×
0.001
0.598
✓
501
=
0.001
998 × (2 ×
10
-9
)
✓
1.002 ×
10
-6
=
0.001
998
✓
1.4328 ×
10
-7
=
0.598
998 × 4,182
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.