Energy Balances — Introduction to Chemical and Biological Engineering (2022)

Rationale

When we learned material balances,we were able to track the movement of chemical species throughout a system orprocess.

Another important aspect of process analysis is the determination of theenergy requirements and temperatures around systems.For instance, how can we maintain a consistent temperature in a reactor if thereaction inside is exothermic? How much energy do we need to add to a processstream to move it to some new temperature?

Important nomenclature

A closed system is one in which there is a fixed volume or space andno streams entering or leaving the system.

An open system is one in which there are streams entering and leaving.These streams can add or remove material and energy from a system.

An adiabatic process is one in which there is no heat added or removedfrom the system.

An isothermal process is one in which the system stays at a constanttemperature. Heat may need to be added or removed to maintain this condition.

First law of thermodynamics for closed systems

\[\Delta E = Q + W\]

where

\(E\)

=

total energy of the system (units of energy)

\(Q\)

=

heat transferred from the environment to the system through the boundaries of the volume (units of energy) over the interval of time during which E changes

\(W\)

=

work done on the system by the environment (units of energy) over the interval of time during which E changes

Typical units for these energy terms are

  • erg and calorie (\(\si{cal}\)) in the cgs system

  • Joule (\(\si{J}\)) in the SI system

  • British thermal unit (\(\si{Btu}\)) in the American system

‘Direction’ of heat and work

If work is done on a system, energy is utilized to providethat work, and the result is an increased energy within the system.

Similarly, if heat passes across a system boundary into a system, it likewiseshows up as increased energy within the system.

Consistent with this idea is the convention in this course that\(Q\) and \(W\)have a positive sign if they add energy to the system.

More symbols we need to know

Recall that a ‘dot’ over a symbol means a rate (quantity per unit time).

For example, \(\dot m\) means a mass per unit time in units such as\(\si{g/s}\).

(Video) Introduction to Energy Balances - Part 1

We now introduce a ‘hat’ over symbols. This representsa quantity per unit mass. This is often called a specific quantity.

For instance \(\hat E\) is the specific energy(or energy per unit mass). Units in this case are something like\(\si{J/g}\).

First law of thermodynamics for open systems

For a steady-state open system, our conservation law becomes

\[\begin{split}\sum_{\substack{\text{output}\\\text{streams}}} (\dot m \, \hat E)_{out} -\sum_{\substack{\text{input}\\\text{streams}}} (\dot m \, \hat E)_{in} =\dot Q + \dot W\end{split}\]

where

\(\dot m\)

=

mass flow rate of a stream (units of mass per time)

\(\hat E\)

=

energy per mass of a stream of flowing material

\(\dot Q\)

=

rate of transfer of energy across the boundaries of a system into that system (units of energy per time)

\(\dot W\)

=

rate that work is done on a system (units of energy per time)

Typical units for these rate of energy transfer terms are

  • erg per second (\(\si{erg/s}\)) andcalorie per second (\(\si{cal/s}\)) in the cgs system

  • Joule per second (i.e., Watts) (\(\si{J/s}=\si{W}\)) in the SI system

  • British thermal unit per hour (\(\si{Btu/hr}\)) in the American system

There are three forms of energy that we will consider (expressed per mass ofmaterial):

  • internal energy

  • kinetic energy

  • potential energy

Just as a flowing stream has kinetic and potential energy,individual molecules also have kinetic energy (from their individual motion)and potential energy (from the attraction and repulsion between molecules).

The sum of these molecular energies is expressed as the internal energy of the material,which is a strong function of temperature.

Now, the total energy per mass of material is

\[\begin{split}\begin{align*}\hat E_{\text{total}} & = \hat E_{\text{internal}} + \hat E_{\text{kinetic}} + \hat E_{\text{potential}} \\ & = \hat U + \frac{1}{2} \, \alpha \, v^{2} + g \, z\end{align*}\end{split}\]

What is \(\alpha\) in the equation above and those to follow?

In many cases of practical interest, the kinetic energy term is moreappropriately written

(Video) Introduction to Energy Balances for Stirred Tank Reactors

\[\hat E_{\text{kinetic}} = \frac{1}{2} \, (v^{2})_{\text{avg}}\]

We can relatively easily measure the average velocity from the volumetricflow rate and cross sectional area, \(v_{\text{avg}} = \dot V / A_{c}\).

Thus, we also easily estimate the average velocity squared,\((v_{\text{avg}})^{2}\).However, our kinetic energy term, \((v^{2})_{\text{avg}}\),is not equal to \((v_{\text{avg}})^{2}\).

To make terms that are equivalent, we define a correction factor,\(\alpha\), such that

\[(v^{2})_{\text{avg}} = \alpha \, (v_{\text{avg}})^{2}\]

Though not strictly correct, in most cases in this course, we will assumethat \(\alpha = 1\).

Our generalized conservation of energy equation is therefore

\[\begin{split}\sum_{\substack{\text{output}\\\text{streams}}} \left [\dot m \left(\hat U + \frac{1}{2}\, \alpha \, v^{2} + g \, z \right) \right ]_{out} -\sum_{\substack{\text{input}\\\text{streams}}} \left[\dot m \left(\hat U + \frac{1}{2}\, \alpha \, v^{2} + g \, z \right) \right]_{in} =\dot Q + \dot W\end{split}\]

Note that the internal energy is often combined with the flow work into a propertycalled the enthalpy, which is represented by the symbol \(\hat H\)and defined as

\[\hat H = \hat U + P \, \hat V\]

To be consistent with other units in our equation, specific enthalpy has unitsof energy per unit mass, e.g., \(\si{Btu/lb_{m}}\) or \(\si{J/g}\).

Rate of Work

When external forces do work on a fluid, the energy of that fluid increases.

For example, when a pump does work on a fluid, that work increases fluidvelocities, potential energy, and/or fluid temperature.Fluids can also do work on their environment, and thereby lose energy.

It is important to note that the rate of work \(\dot W\) consists of both‘rate of flow’ work and shaft work: \(\dot W = \dot W_{PV} + \dot W_{S}\).

Rate of flow work, \(\dot W_{PV}\):

This work results from the displacement of fluid during flow and is similar tothe pressure-volume work associated with the compression or expansion of aclosed system.However, in the case of an open system (with inlet and outlet streams), theflow of fluid into and out of a system represents a continual performance ofwork as upstream fluid “pushes” fluid into the system entrance and the fluidin the system “pushes” downstream fluid out of the system exit.

Rate of shaft work, \(\dot W_{S}\):

When the flowing fluid in a system contacts moving parts, work is performed.Shaft work is positive when the net work is on the system(such as in a pump, where the moving parts are driven by external forces andthereby “push” the fluid). Conversely, the shaft work has a negative valuewhen the net work is done by the fluid (such as in a turbine, where the fluidcauses the parts to move).

Utilizing the above relationships and the fact that\(P \, \hat V = P/\rho\),we obtain the common form of the conservation of energy equation:

Conservation of energy equation

\[\begin{split}\sum_{\substack{\text{output}\\\text{streams}}} \left [\dot m \left(\hat H + \frac{1}{2}\, \alpha \, v^{2} + g \, z \right) \right ]_{out} -\sum_{\substack{\text{input}\\\text{streams}}} \left[\dot m \left(\hat H + \frac{1}{2}\, \alpha \, v^{2} + g \, z \right) \right]_{in} =\dot Q + \dot W_{S}\end{split}\]

Special cases

For now, we’ll focus on applications of the steady-state energy balance in whichthere is negligible change in kinetic and potential energies andno shaft work.

Common form of the conservation of energy equation

\[\begin{split}\sum_{\substack{\text{output}\\\text{streams}}} \left( \dot m \,\hat H \right )_{out} -\sum_{\substack{\text{input}\\\text{streams}}} \left( \dot m \,\hat H \right )_{in} =\dot Q\end{split}\]

Sensible heating or cooling

When a material is warmed or cooled without a phase change, we call this processsensible heating or cooling.

For sensible heating/cooling, the specific enthalpy, \(\hat H\),can be approximated as

\[\hat H \approx \bar C_{p} \, (T - T_{\text{ref}})\]

where \(\bar C_{p}\) is the heat capacity averaged from the referencetemperature \(T_{\text{ref}}\) to the temperature of interest, \(T\).

The units of \(\bar C_{p}\) are energy per mass per temperature change,such as \(\si{cal/g.\degree C}\) or \(\si{Btu/lbm.\degree F}\).

Energy balance for sensible heating or cooling

(Video) Introduction to Energy Balances for Plug Flow Reactors

\[\begin{split}\sum_{\substack{\text{output}\\\text{streams}}} \left[ \dot m \, \bar C_{p} \, (T - T_{\text{ref}}) \right ]_{out} -\sum_{\substack{\text{input}\\\text{streams}}} \left[ \dot m \, \bar C_{p} \, (T - T_{\text{ref}}) \right ]_{in} =\dot Q\end{split}\]

Exercise: Energy balance for sensible heating or cooling

Consider a process designed to process biomass for further processinginto biofuels.

Stream

Mass flow rate

Heat capacity

Temperature

(\(\si{kg/min}\))

(\(\si{J/g\,K}\))

(\(\si{\degree C}\))

raw biomass

\(15\)

\(0.9\)

\(90\)

pure water

\(120\)

\(4.2\)

?

diluted biomass

\(45\)

\(2.6\)

\(60\)

contaminated water

?

\(3.5\)

\(80\)

1. If the process is adiabatic, what temperature of pure water is requiredto produce diluted biomass at a temperature of \(\SI{60}{\degree C}\)?Assume a reference temperature of \(\SI{25}{\degree C}\).

2. If the temperature of the pure water is the same as in part a,how much heat (\(\si{W}=\si{J/s}\)) must be added or removed from theprocess to lower the diluted biomass temperature an additional\(\SI{10}{\degree C}\)?

Answers

Latent (phase change) heating or cooling

When a material changes phase (water to steam, water to ice) without a changein temperature, we call this process latent heating or latent cooling.

Considering phase changes with only one inlet stream and one outlet streamand where the phase change occurs isothermally at the temperature for whichwe have a value for \(\Delta H_{\text{phase change}}\), we can write

\[\hat H_{\text{out}} - \hat H_{\text{in}} = \Delta \hat H_{\text{phase change}}\]

(Video) Energy Balance on a Single-Phase System

This relationship, when combined with our energy balances, leads to

Steady-state energy balance for phase change

\[\dot m_{\text{phase change}} \, \Delta \hat H_{\text{phase change}} = \dot Q_{\text{phase change}}\]

Chemical reactions

\[\begin{split}\begin{align*}\sum_{\substack{\text{output}\\\text{streams}}} \left( \dot m \, \hat H \right )_{out} -\sum_{\substack{\text{input}\\\text{streams}}} \left( \dot m \, \hat H \right )_{in} &= \left ( \frac{\text{moles A reacted}}{\text{time}} \right ) \Delta \tilde H_{\text{reaction, A}} \\ &= r_{\text{consumption, A}} \, \Delta \tilde H_{\text{reaction, A}}\end{align*}\end{split}\]

Steady-state energy balance for systems with chemical reactions

\[r_{\text{consumption, A}} \, \Delta \tilde H_{\text{reaction, A}} = \dot Q_{\text{reaction}}\]

Exercise: Energy balance for a system with a chemical reaction

Suppose the following reaction is carried out in a chemical reactor:\(\ce{A + B -> C}\).

The reactor has a single inlet and a single effluent (outlet) and the entirereactor system is at constant density (\(\rho = \SI{0.9}{kg/L}\)).

The desired conversion of \(A\) is \(0.8\).

Operating conditions and parameter values

  • feed conditions:
    • \(\dot V_{\text{feed}} = \SI{50}{L/hr}\)

    • \(c_{\text{A, feed}} = \SI{1}{M}\)

    • \(T_{\text{feed}} = \SI{50}{\degree C}\)

  • heat (enthalpy) of reaction:
    • \(\Delta \tilde H_{\text{reaction, A}} = \SI{−200}{kJ/\text{mol of A}}\)

  • heat capacities:
    • \(C_{\text{p, feed}} = \SI{1.7}{kJ/kg.K}\)

    • \(C_{\text{p, outlet}} = \SI{2.1}{kJ/kg.K}\)

1. Assuming that the reactor is perfectly insulated (adiabatic), whatwould be the temperature of the effluent stream?

2. If we want the system to act isothermally and have the temperature ofthe effluent stream equal to that of the inlet stream, how much energy pertime (in \(\si{Watts}=\si{J/s}\)) would we have to add or remove from thesystem?

Solution

Procedure for using the energy balance

Similar to those used in material balances, here are the recommendedsteps in solving problems in which energy balances are relevant:

  1. Draw a diagram if one is not already available.

  2. Write all known quantities (flow rates, densities, etc.) in theappropriate locations on the diagram.

  3. Identify and assign symbols to all unknown quantities and write them inthe appropriate locations on the diagram.

  4. Write the appropriate simplified energy balance depending on whether theproblem involves sensible heating/cooling, phase change, or chemicalreaction. Along with the balance equation, write down the giveninformation associated with that equation, such as average heatcapacities, enthalpy changes for a phase change. or enthalpy changes ofreaction.

  5. Construct appropriate material balance equations to aid in determiningunknown flow rates or other material-re lated information.Continue to seek such equations, as needed, until the total number ofequations equals the number of unknowns.

  6. Solve the equations to determine the desired unknown quantities.

FAQs

How do you calculate energy balance in chemical engineering? ›

How to do an energy balance in the PRESENCE of - YouTube

Why is energy balance important in chemical engineering? ›

Energy balances are used in the examination of the various stages of a process, over the whole process and even extending over the total production sys- tem from the raw material to the finished product.

What is Q in energy balance? ›

1. Heat (Q) – energy that flows due to a temperature difference. between the system and its surroundings. – always flows from high to low temperature.

How do you calculate the energy balance of a reactor? ›

  1. The energy balance for the constant-pressure case follows from Equation 6.15. CP.
  2. dT. dt= −∆HRknA.
  3. in which CP = VRρ ˆ
  4. CP is the total constant-pressure heat capacity. For an ideal gas, we. know from thermodynamics that the two total heat capacities are simply related,
  5. CV = CP − nR. (6.21)

What is the chemical formula for energy? ›

E = w + q.

What is energy balance method? ›

The energy balance method is a tool that permits to directly predict resonances for a conservative system with nonlinearities from its nonlinear modes.

Why energy balance is important? ›

Energy balance is important. When you consume too much energy and burn too little, your body stores that excess energy as body fat. And being overweight increases your risk for several cancers, including colon, pancreatic, endometrial and post-menopausal breast cancer.

What is the importance of energy balance in process control? ›

Material and Energy balances are important, since they make it possible to identify and quanti- fy previously unknown losses and emissions. These balances are also useful for monitoring the improvements made in an ongoing project, while evaluating cost benefits.

What is the purpose of material and energy balance? ›

The basic purpose of material and energy balance is • to quantify all the material, energy and waste streams in a process or a system. to find out the difference between calculated/designed values and measured/actual values thereby making it possible to identify previously unknown losses and emissions.

What is H in energy balance equation? ›

Note that the internal energy is often combined with the flow work into a property called the enthalpy, which is represented by the symbol and defined as. H ^ = U ^ + P V ^

What is heat duty of reactor? ›

In case of specified heat duty, the reactor temperature is calculated from the energy balance; the energy balance is based on the enthalpy that includes formation heats and does not require specification of heats of reaction. The heat duty can be specified directly or via an energy inlet stream.

How do you calculate heat balance? ›

Heat balance (thermal equilibrium) is the balance between the rate of heat production and the rate of heat loss.
  1. Heat balance equation.
  2. Heat production = the rate of heat production = M - W where: ...
  3. Heat loss = the rate of heat loss = R + C + E + L + K + S where: ...
  4. Heat exchange mechanisms - 3 main physiological mechanisms.

How do you calculate the energy balance for a distillation column? ›

Energy balance example - YouTube

How do you calculate reactor capacity? ›

Using bulk density and weight loaded in the reactor you can calculate the catalyst volume using density=mass/volume formula.

What is the principle of material balance? ›

Material balance is based on the mass conservation principle which states that the sum of the weight of all inputs must be exactly equal to the sum of all outputs.

What is energy balance engineering? ›

(Chemical Engineering: General) An energy balance is a consideration of the energy input, output, and consumption or generation in a process or stage. In establishing an energy balance, all sources of thermal energy are put on the input side, and all items of heat utilization on the output side.

How do you calculate heat balance? ›

Heat balance (thermal equilibrium) is the balance between the rate of heat production and the rate of heat loss.
  1. Heat balance equation.
  2. Heat production = the rate of heat production = M - W where: ...
  3. Heat loss = the rate of heat loss = R + C + E + L + K + S where: ...
  4. Heat exchange mechanisms - 3 main physiological mechanisms.

What is energy balance in thermodynamics? ›

The balance is expressed in words as: all energies into the system are equal to all energies leaving the system plus the change in storage of energies within the system.

What is mass and energy balance? ›

A mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique.

Videos

1. Introduction to Energy
(LearnChemE)
2. What is Chemical and Biological Engineering?
(Iowa State University College of Engineering)
3. Material and Energy Balances
(NPTEL-NOC IITM)
4. Energy Balance on a Condenser
(LearnChemE)
5. Concept in Chemical Engineering - Heat and Energy Balance
(Penn State College of Engineering)
6. PFR Energy Balances
(LearnChemE)

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