Consider a sample of disc-shaped (cylindrical) particles with circular cross-section of varying diameter D. All particles have the same thickness L equal to 1 um. The sample contains 90% (by volume) of particles with diameter 100 m, 9% (by volume) of particles with diameter 10 um and 1% (by volume) of particles with diameter 1 um. (a) Calculate the number fraction of particles of each diameter in the sample. (b) Determine the volume equivalent sphere diameter dv of particles of each diameter. (c) Calculate the number weighted mean, dv,N, and the volume weighted mean, dv,V, of the volume equivalent sphere diameter of particles in the sample. (d) What volume fractions of each of the three sizes would you choose in order to get a sample with a number weighted mean, dv,N, equal to 10 um? um = 10-6 m
Consider a sample of disc-shaped (cylindrical) particles with circular cross-section of varying diameter D. All particles have the same thickness L equal to 1 um. The sample contains 90% (by volume) of particles with diameter 100 m, 9% (by volume) of particles with diameter 10 um and 1% (by volume) of particles with diameter 1 um. (a) Calculate the number fraction of particles of each diameter in the sample. (b) Determine the volume equivalent sphere diameter dv of particles of each diameter. (c) Calculate the number weighted mean, dv,N, and the volume weighted mean, dv,V, of the volume equivalent sphere diameter of particles in the sample. (d) What volume fractions of each of the three sizes would you choose in order to get a sample with a number weighted mean, dv,N, equal to 10 um? um = 10-6 m
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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Consider a sample of disc-shaped (cylindrical) particles with circular cross-section of varying diameter D. All particles have the same thickness L equal to 1 um. The sample contains 90% (by volume) of particles with diameter 100 m, 9% (by volume) of particles with diameter 10 um and 1% (by volume) of particles with diameter 1 um. (a) Calculate the number fraction of particles of each diameter in the sample. (b) Determine the volume equivalent sphere diameter dv of particles of each diameter. (c) Calculate the number weighted mean, dv,N, and the volume weighted mean, dv,V, of the volume equivalent sphere diameter of particles in the sample. (d) What volume fractions of each of the three sizes would you choose in order to get a sample with a number weighted mean, dv,N, equal to 10 um?
um = 10-6 m
answers attached show workiong on how to reach these
Transcribed Image Text:Relative frequency: f₁ = f(x)x₁<x²x₁ = Q(x;) — Q(x₁_) = | q(x)dx
XH-1
Cumulative frequency distribution: Q(x) =
Differential frequency distribution: q (x) =
Weighted mean size: x₁ =
[w, x,
j=1
Σw,
W₁
j=1
Total volume of all particles:
Number weighted mean size: μ₁/Mo
Length weighted mean size: μ₂/M₁
Surface weighted mean size: μ3/μ₂
Volume weighted mean size: 4/H3
St =
= 1=1
Amount of particles with size ≤ x
Amount of all particles
q(x₁) ≈
N,w;x;
EN,w,
i=1
St=
dQ(x)
dx
Nis
n-th moment of particle size distribution: µ„ = = [ƒ‚x" = √x"q„(x)dx
i=1
or
fwx,
ΣΥw
fm,
1=1
x
Nbins
V₂ = - Σ Nβx = βN [x³ qn(x) dx
i=1
Total surface area of all particles (for cube and sphere shapes):
Nbins
ΣN₁αx² = aN
i=1
[w(x) xq (x) dx
Tw(x) q, (x) dx
= X
Nfx
Q(x₁+1)-Q(xi
Xi+1-Xi
x² qn(x) dx
Total surface area of all particles (for cylinder shapes with constant L):
Nbins
Σ N₁ (²nd² + πd₁L) = N
i=1
x² qn(x) dx + TL x qn(x) dx
"S
da
Total surface area of all particles (for cylinder shapes with constant d):
Nbins
S₁ = [N, (²nd² + nd²₁) = N( ²
i=1
d²
= N ( ½ d² + m² [ x ₁₂ (x) dx )
πα
Transcribed Image Text:(a) 0.00819; 0.0819; 0.9099
(b) dv = 24.7, 5.31 and 1.14 µm
(c) d1,0 = 1.67 μm; d4,3 = 22.7 µm
(d) There are many different solutions (for example, 1 = 0.9696, 2 = 0.0304, 03 = 0)
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Step 1: Determine(a)Number of fraction of particles(b)Volume equivalent diameter(c)Weighted mean,volume mean
VIEWStep 2: Calculate the number of fraction particles of each diameter of sample.
VIEWStep 3: Calculate volume equivalent sphere diameter dv of each diameter.
VIEWStep 4: Calculate weighted mean and volume weighted mean.
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