Conservation of MassME121, Portland State University

Mass Balance

• “Mass can neither be created nor destroyed… just rearranged”

• In fluid dynamics, chemistry, heat transfer, etc:

• min + mgenerated - mout - mconsumed = Δmsystem

with Δmsystem, accumulation of mass in the system

Example applications• Water in lake:

mwater flowing in + rain – mwater flowing out - evaporation = Δmsystem

• Water in fishtank:

minitally in tank+ m added from reservoirs– mleaving through overflow = mfinal in tank

with: • minitally in tank = mfinal in tank

• m added from reservoirs= mleaving through overflow

Class Problem

A 20L aquarium contains 2% salt by weight. How much salt would you need to add to bring the salt concentration to 3.5% salt by weight?

1. Draw a diagram to present the system

2. Label all inputs and outputs, assigning variables to unknowns

min=?

XSin=2/100

Vin=20L

mout=?

XSout=3.5/100

Vout≠ Vin

mSadd=?

XSadd = 1

3. Apply conservation of mass to each component in play (water and salt) and for the mixture:

• Overall: min + madd = mout

• Salt: mSin + mSadd = mSout <-> XSin*min+ XSadd*madd = XSout*mout

• Water: mWin + mWadd = mWout <-> XWin*min+ XWadd*madd = XWout*mout

with:

XSin+ XWin = 1

XSadd+ XWadd = 1

XSout+ XWout = 1

min=?

XSin=2/100

Vin=20L

mout=?

XSout=3.5/100

Vout≠ Vin

mSadd=?

XSadd = 1

4. Convert any volume to mass when possible:

min = Vin * ρin

with ρin, density of water (assume ρin = 1 kg/L if not specified)

min = 20 L * 1 kg / L = 20 kg

min=?

XSin=2/100

Vin=20L

mout=?

XSout=3.5/100

Vout≠ Vin

mSadd=?

XSadd = 1

5. Solve for mass balance:

• Overall: min + madd = mout

20 kg + madd = mout

• Salt: mSin + mSadd = mSout <-> XSin*min+ XSadd*madd = XSout*mout

0.02*20kg + 1*madd = 0.035*mout

0.4 kg + madd = 0.035*mout

2 equations, 2 unknowns: we can solve

min=?

XSin=2/100

Vin=20L

mout=?

XSout=3.5/100

Vout≠ Vin

mSadd=?

XSadd = 1

mout = 20 + madd

madd - 0.035madd = 0.7 – 0.4

mout = 20 + madd

madd (1-0.035) = 0.3

mout = 20 + madd

madd = 0.311 kg = mSadd

min=?

XSin=2/100

Vin=20L

mout=?

XSout=3.5/100

Vout≠ Vin

mSadd=?

XSadd = 1

20 + madd = mout

0.4 + madd = 0.035*mout

mout = 20 + madd

0.4 + madd = 0.035*(20 + madd )

mout = 20 + madd

0.4 + madd = 0.7 + 0.035madd