Summary 3: Enthalpy and the Energy Balance

Energy Balance Diagram

\[\text{Accumulation of Energy} = \text{Energy In} - \text{Energy Out}\]

\[\frac{dE_{sys}}{dt} = \dot{E}_{in} + \dot{Q} + \dot{W} - \dot{E}_{out}\]

Energy accumulation in a system

\[\frac{dE_{sys}}{dt} = \dot{E}_{in} + \dot{Q} + \dot{W} - \dot{E}_{out}\]

Energy can be accumulated in a system as internal energy, kinetic energy, or potential energy.

Therefore, the total energy of a system is: \[E_{sys} = U + E_k + E_p\]

Energy entering and leaving a system

Energy can enter or leave a system within a mass flow as internal energy, kinetic energy or potential energy.

Therefore: \[\dot{E}_{in} = \dot{U}_{in} + \dot{E}_{k,in} + \dot{E}_{p,in}\] \[\dot{E}_{out} = \dot{U}_{out} + \dot{E}_{k,out} + \dot{E}_{p,out}\]

Shifting mass in/out of the system

Work is needed when:

• mass enters or leaves a system
• a system expands or compresses

Must account for this work in the energy balance

When mass enters a system, internal energy and work is added.

Enthalpy

The extra work can by accounted for by using enthalpy instead of internal energy

Enthalpy is defined as: \[H = U + PV\]

Key Term

Enthalpy (\(H\)) — a convenient state function defined by \(H=U+PV\).

Combining all the terms

\[\frac{d(m_{sys}H_{sys})}{dt} = \dot{m}_{in} H_{in} - \dot{m}_{out} H_{out} + \dot{Q} + \dot{W}\]

• \(m_{sys}\) is the mass of the system (kg)
• \(H_{sys}\) is the enthalpy of the system (J/kg)
• \(\dot{m}_{in}\), \(\dot{m}_{out}\) are the mass flow rates in or out of the system (kg/s)
• \(H_{in}\), \(H_{out}\) are the enthalpy of the streams in or out of the system (J/kg)
• \(\dot{Q}\) is the rate of heat transfer to the system (W)
• \(\dot{W}\) is the rate of work done on the system (W)

Multiple streams

For multiple stream, include each term:

\[ \begin{aligned} \frac{d(m_{sys}H_{sys})}{dt} ={} &\dot{m}_{in,1} H_{in,1} + \dot{m}_{in,2} H_{in,2} + \dot{m}_{in,3} H_{in,3} \\ & - \dot{m}_{out,1} H_{out,1} - \dot{m}_{out,2} H_{out,2}\\ & + \dot{Q}_1 + \dot{Q}_2 - \dot{Q}_3\\ & + \dot{W} \end{aligned} \]

The General Energy Balance Equation

The general energy balance equation can describe any process:

\[\frac{d(m_{sys}H_{sys})}{dt} = \sum_i \dot{m}_{i,i} H_{in,i} - \sum_j \dot{m}_{out,j} H_{out,j} + \sum_k \dot{Q}_k + \sum_l \dot{W}_l\]

The summation terms allow for multiple:

• flow streams into the system
• flow streams out of the system
• heat and work transfer terms over the system-surrounding boundary

The General Energy Balance Equation

If the system is described in moles rather than mass:

\[\frac{d(n_{sys}H_{sys})}{dt} = \sum_i \dot{n}_{i,i} H_{in,i} - \sum_j \dot{n}_{out,j} H_{out,j} + \sum_k \dot{Q}_k + \sum_l \dot{W}_l\]

where:

• \(n_{sys}\) is the moles of system
• \(\dot{n}\) is the molar flow rate in or out of the system

Sometimes mixed units are used. Be careful !

Example 3.1: Identifying the terms in the energy balance

For each system below, start from the general energy balance and simplify. State the assumption that lets you remove each term — don’t delete terms from memory.

\[\frac{d(m_{sys}H_{sys})}{dt} = \sum_i \dot{m}_{in,i} H_{in,i} - \sum_j \dot{m}_{out,j} H_{out,j} + \sum_k \dot{Q}_k + \sum_l \dot{W}_l\]

(a) A heat exchanger (steady state). A hot stream and a cold stream pass through a heat exchanger and swap heat. Write the balance for two different choices of boundary:

• 1. the boundary drawn around the entire exchanger (both streams inside), and
• 2. the boundary drawn around the hot side and the cold side separately.

(b) An insulated tank (closed). Ice and liquid water are sealed in a well-insulated tank and left to reach equilibrium as the ice melts. Nothing flows in or out.

(c) A pump feeding a steam-injection point (steady state). Liquid is pumped up to pressure, then steam is injected into the pumped stream to heat it. Take the system as the pump plus the injection point.