Source: ENGR 216 Comprehensive Exam Practice Bank
Tags: mass balance, universal accounting equation, UAE, conservation of mass, mixing tank, concentration, mass flow rate, steady state, ENGR 216, Texas A&M
The universal accounting equation (UAE) is the conservation-of-mass principle applied to engineering systems: what goes in, minus what goes out, plus what is generated, minus what is consumed, equals the accumulation. For simple mixing problems at steady state, this reduces to balancing the mass flow rates and concentrations of each component across all inlets and outlets.
Universal accounting equation (UAE)
$$\text{In} - \text{Out} + \text{Generation} - \text{Consumption} = \text{Accumulation}$$
The general form of conservation applied to mass, energy, or any extensive property.
Mass balance
A specific application of the UAE to mass. For systems with no chemical reaction and at steady state, generation, consumption, and accumulation are all zero, leaving $\text{In} = \text{Out}$.
Steady state
A condition where nothing in the system changes with time. All accumulation terms are zero.
Mass flow rate ($\dot{m}$)
The mass of material entering or leaving a system per unit time, typically in kg/min or kg/s.
Concentration (mass fraction)
The proportion of a specific component in a mixture, expressed as a decimal or percentage. For sugar in juice, a 29% sugar concentration means 0.29 kg of sugar per kg of juice.
Continuously stirred tank
A well-mixed vessel where the outlet composition equals the composition inside the tank at any instant. This assumption simplifies the mass balance because you do not need to model spatial variation.
For a steady-state mixing problem with no reactions:
Overall mass balance: The total mass flow in must equal the total mass flow out.
Component mass balance: The total mass flow of a specific component in must equal the total mass flow of that component out.
When the tank is accumulating or depleting, the overall balance becomes:
$$\sum \dot{m}{\text{in}} - \sum \dot{m}{\text{out}} = \frac{dm_{\text{tank}}}{dt}$$
But at steady state, $dm/dt = 0$, so total in = total out.
Setup:
Input 1: Apple juice, 29% sugar, flow rate 10 kg/min
Input 2: Orange juice, 15% sugar, flow rate 18 kg/min
Output: Mixed juice drained at 20 kg/min
Step 1: Overall mass balance
Total in = 10 + 18 = 28 kg/min. Total out = 20 kg/min. Since 28 > 20, mass is accumulating in the tank at 8 kg/min. The tank is not at steady state overall, but the outlet composition still comes from the well-mixed contents.
Step 2: Component (sugar) mass balance
Sugar in from apple juice: $0.29 \times 10 = 2.90$ kg/min. Sugar in from orange juice: $0.15 \times 18 = 2.70$ kg/min. Total sugar in = 5.60 kg/min.
Step 3: Determine outlet concentration
For a continuously stirred tank, the outlet concentration equals the instantaneous tank concentration. The tank concentration is governed by the ratio of total sugar input to total mass input (because both sugar and total mass accumulate proportionally in a well-mixed tank):
$$C_{\text{sugar}} = \frac{\text{Total sugar in}}{\text{Total mass in}} = \frac{5.60}{28} = 0.20 = 20.00%$$
The outlet drains at this concentration. The answer is 20.00.
A frequent mistake is dividing the sugar mass flow by the outlet flow rate (20 kg/min) instead of the total inlet flow rate (28 kg/min). The continuously stirred tank assumption means the concentration inside the tank (and therefore at the outlet) is set by the ratio of inputs, not by the drain rate.
Formula | Use |
|---|---|
$\text{In} - \text{Out} + \text{Gen} - \text{Con} = \text{Acc}$ | Universal accounting equation |
$\sum \dot{m}{\text{in}} = \sum \dot{m}{\text{out}}$ | Steady-state mass balance (no reaction) |
$C_{\text{out}} = \frac{\sum (\dot{m}_i \times C_i)}{\sum \dot{m}_i}$ | Outlet concentration from a well-mixed tank |
⚠️ Always check whether the system is at steady state. If total mass in does not equal total mass out, there is accumulation.
⚠️ For a continuously stirred tank, outlet concentration equals the concentration inside the tank, which equals total component inflow divided by total mass inflow.
⚠️ Do not divide the component mass flow by the outlet flow rate. Divide by total inlet flow rate to get the tank concentration.
⚠️ Concentration can be given as a percentage or a decimal. Be consistent, and convert early.
⚠️ The UAE with generation and consumption terms appears when chemical reactions are involved. For simple mixing (no reaction), set those terms to zero.
Q: Apple juice (29% sugar, 10 kg/min) and orange juice (15% sugar, 18 kg/min) are mixed in a stirred tank. The outlet drains at 20 kg/min. What is the sugar concentration of the mixture?
A: Sugar in = (0.29 × 10) + (0.15 × 18) = 2.90 + 2.70 = 5.60 kg/min. Total mass in = 28 kg/min. Concentration = 5.60 / 28 = 0.20, or 20.00%.
Q: In the UAE, what does each term represent?
A: In = mass entering the system. Out = mass leaving. Generation = mass created by reaction. Consumption = mass destroyed by reaction. Accumulation = change in mass stored within the system.
Q: Why do you divide by total inlet flow rather than outlet flow to find concentration?
A: In a continuously stirred tank, the concentration inside the tank (and at the outlet) is determined by the composition of all material entering the tank, not by how fast material is drawn off. The drain rate affects accumulation, not composition.
mass balance, universal accounting equation, UAE, conservation of mass, mixing, stirred tank, CSTR, continuously stirred, mass flow rate, concentration, mass fraction, sugar concentration, steady state, accumulation, generation, consumption, inlet, outlet, ENGR 216, Texas A&M