Energy Balance for Continuous Flow Thermo

Physical fundamentals

Howard D. Goodfellow , Eric F. Curd , in Industrial Ventilation Design Guidebook (Second Edition), 2020

4.2.2 Fundamentals

Air is seldom dry; it normally contains varying amounts of moisture. Humid air is a mixture of dry air and water vapor. The term dry air denotes the mixture of all gases present in the air (nitrogen, oxygen, carbon monoxide, and inert gases), except water vapor.

The molar mass of dry air is dependent on the consistency of air, but for standard air, it is M _i =0.028964   kg/mol. In practical calculations, we may use the rough value of 0.0290   kg/mol. The molar mass of water vapor is M _h =0.0180153   kg/mol, and the rough value used in practical calculations is 0.0180   kg/mol.

According to the state equation of ideal gas, the partial density of dry air in humid air is

(4.76) $p_i=\frac{p_i{M}_i}{RT},$

where p _i is the partial pressure of dry air and R is the gas constant. According to present knowledge, the best value for the gas constant is

R=(8.31441±0.00026)

In practical calculations, we use the value of R=8.314   J/(mol   K).

The partial density of water vapor in humid air is

(4.77) ${\rho }_h=\frac{p_h{M}_h}{RT},$

where p _h is the partial pressure of water vapor.

The density of humid air is the sum of the partial densities of dry air and water vapor

(4.78) $\rho ={\rho }_i+{\rho }_h$

and the total pressure of humid air is a sum of the partial pressures of dry air and water vapor

(4.79) $p=p_i+p_h$

The mass of dry air in a volume V is denoted as m _i (ρ _i =m _i /V _i ), and the mass of water vapor in V is m _h (ρ _h =m _h /V _h ). Humidity of air, meaning the ratio of water vapor mass to dry air mass, is defined as

(4.80) $p_h=m_h/V_h$

(4.81) $x\equiv \frac{m_h}{m_i}.$

Using partial pressures, this definition can be written as

(4.82) $x\equiv \frac{{\rho }_h}{{\rho }_i}.$

Humidity x is thus a dimensionless number, but sometimes a "dimension" is added to it, as a reminder of its definition. We can, for instance, write x=0.05 or x=0.05   kg H₂O/kg d.a., where d.a. stands for dry air.

According to Eqs. (4.76), (4.78), and (4.80), humidity can be written as

(4.83) $x=\frac{M_h}{M_i}\frac{p_h}{p_i}=0.6220\frac{p_h}{p_i}=0.6220\frac{p_h}{p-p_h}.$

By solving the partial pressure of water vapor from the earlier equation, we receive

(4.84) $p_h=\frac{xp}{0.6220+x}.$

We denote again the mass of dry air in a volume V as m _i and the mass of water vapor as m _h . When humid air is treated as an ideal mixture of two components, dry air and water vapor, the enthalpy of this mixture is

(4.85) $H=m_i{h}_i+m_h{h}_h,$

where h _i is the specific enthalpy of dry air (J/kg), and h _h is the specific enthalpy of water vapor.

Technical calculations dealing with humid air are reasonable to solve with dry air mass flow rates, because these remain constant in spite of changes in the amount of water vapor in the air. For that reason a definition for enthalpy,

(4.86) $h_k=\frac{H}{m_i},$

which is the enthalpy of humid air divided by the dry air mass, is made. The dimension of enthalpy h _k is J/kg, but it is often written as J/kg d.a. as a reminder that the total enthalpy of the mixture is calculated in terms of a kilogram of dry air.

Combining Eqs. (4.85) and (4.86), we get

$m_i{h}_k=m_i{h}_i+m_h{h}_h,$

and using Eq. (4.81), we have

(4.87) $h_k=h_i+x{h}_h.$

In calculations with humid air, when the pressure is not high (usually the atmospheric pressure of 1   bar), water vapor and dry air can be handled as an ideal gas, as we have already done in Eqs. (4.76) and (4.78). For ideal gases the specific enthalpy is just a function of temperature

$h_i=h_i\left(T\right)$

and

$h_h=h_h\left(T\right).$

When 0°C dry air is chosen as the zero point of dry air enthalpy, and 0°C water as the zero point of water vapor enthalpy, the enthalpies of dry air and water vapor can be calculated from the equations

(4.88) $h_i(T)={\int }_{273.15\mathrm{K}}^T{c}_{pi}(T)dT$

(4.89) $h_h(T)=l_{ho}+{\int }_{273.15\mathrm{K}}^T{c}_{ph}(T)dT$

where c _pi (T) is the specific heat of dry air [J/(kg   K)], c _ph (T) is the specific heat of water vapor, and l _ho is the heat of vaporization at 0°C. Its numerical value is

$l_{ho}=2501\text{kJ}/\text{kg}$

Specific heats c _pi and c _ph are somewhat dependent on temperature. In the temperature range of −10°C to +40°C, their average values are

$c_{pi}=1.006\text{kJ}/(\text{kg}°C)$

$c_{ph}=1.85\text{kJ}/(\text{kg}°C).$

At the temperature of +50°C, their values are c _pi =1.008   kJ/(kg   °C) and c _ph =1.87   kJ/(kg   °C).

Using numerical values mentioned earlier, the enthalpy of humid air h _k (Eq. 4.87) can be written as

(4.90a) $h_k=1.006\theta +x\left(2501+1.85\theta \right)\mathrm{kJ}/\mathrm{kg},$

where θ is the temperature in Celsius. Eq. (4.90a) can also be written as

(4.90b) $h=c_{pi}\theta +x\left(l_{ho}+c_{ph}\theta \right)$

We denoted the mass of dry air in a volume V as m _i , that is, ρ _i =m _i /V _i , and the mass of water vapor in V as m _h , that is, ρ _h =m _h /V _h . In practical calculations, we usually handle volume flow q _v (m³/s) instead of volume V. For instance, the value of volume flow is known in the suction inlet of a fan when the operating point of the fan is defined. Volume flow q _v , expressing the total airflow or the combined volume flow of water vapor and dry air, is not constant in various parts of the duct, because the pressure and temperature can vary. Therefore in technical calculations dealing with humid air, material flows are treated as mass flows. Also, while the humidity can vary, the basic quantity is dry air mass flow ṁ _i (kg d.a./s). If, for instance, we know the volume flow q _v of a fan, the dry air mass flow through the fan is

(4.91) ${\dot{m}}_i=p_i{q}_{\upsilon }$

where p _i is the partial pressure of dry air in the suction inlet of the fan, in the same place where the total volume flow $\dot{V}$ is defined. Accordingly, the water vapor flow ṁ _h (kg H₂O/s) along the volume flow is

(4.92) ${\dot{m}}_h=p_h{q}_{\upsilon },$

where p _h is the partial pressure of water vapor.

Due to the definition of humidity (4.87), on the basis of the Eqs. (4.91) and (4.92),

(4.93) ${\dot{m}}_h=x{\dot{m}}_i,$

When an energy balance is written, an enthalpy flow of humid air Ḣ is needed. This can be written according to the Eqs. (4.85) and (4.93) as

$\dot{H}={\dot{m}}_i{h}_i+{\dot{m}}_h{h}_h={\dot{m}}_i(h_i+x{h}_h)$

or briefly, using definition (4.86),

(4.94) $\dot{H}={\dot{m}}_i{h}_k.$

In energy balance calculations, which we will handle later on, we use Eq. (4.94).

Example 1

A fan takes +15°C humid air at 0.5   m³/s. What is the dry air mass flow ṁ _i taken by the fan when the outdoor humidity is x=0.009 and the outdoor pressure is p=1.0   bar?

Partial pressure of water vapor is calculated from Eq. (4.84)

$p_h=\frac{0.009}{0.6220+0.009}\cdot 1.0=0.01426\mathrm{bar}.$

Dry air partial pressure is

$p_i=p–p_h=1.0–0.01426=0.986\mathrm{bar}=0.986{10}^5\mathrm{Pa}.$

Partial density of dry air is calculated from Eq. (4.76)

$p_i=\frac{0.986\cdot {10}^5\cdot 0.0290}{8.314\cdot (273.15+15)}=1.194\mathrm{kg}\mathrm{d}\mathrm{.a}{./m}^3.$

Dry air mass flow is calculated from Eq. (4.91)

${\dot{m}}_i=1.194\cdot 0.5=0.597\mathrm{kg}\mathrm{d}\mathrm{.a}./s.$

Water vapor flow through the fan is

${\dot{m}}_h=x{\dot{m}}_i=0.009\cdot 0.597=0.005373\mathrm{kg}{\mathrm{H}}_{2}O/s.$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128167809000046

Sustainable Energy Technologies & Sustainable Chemical Processes

Mohammad G. Rasul , in Encyclopedia of Sustainable Technologies, 2017

Mass and Energy Balance

A basic mass and energy balance calculation was completed using the experimental results obtained from the analysis of the feedstock and biochars. The mass and energy balance results are shown in Table 9 for MGW. The energy balances show that 41% of the energy in the green waste feedstock was transferred into syngas during the pyrolysis processing. Thus each dry ton of green waste will produce 7.5 GJ of energy. If greater energy production is required (more syngas), the biochar could be partially or completely gasified. For a standard 4   tons/h (dry basis) pyrolysis plant operating on MGW, syngas with an energy value of 30 GJ/h or 8.3   MW would be produced. Conservatively 30%–35% of this syngas energy is required to operate the pyrolysis plant, with the remainder available for thermal or electrical power generation. If the syngas available for export was used in a gas engine for electricity production, a conversion efficiency of 37% might be assumed. This would result in an electrical output of around 2   MW. If the plant was operated over 8000   h a year, on this feedstock it would generate 16,000   MWh per annum.

Table 9. Green waste mass and energy balance

Calculation basis	1000	kg
Biochar yield	41%
	Feed	Biochar
Calorific value (MJ/kg)	18.11	25.86
Mass	1000	410
Energy value (MJ)	18,110	10,603
Energy to syngas (MJ)	7507
	41%

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780124095489101095

Solar–wind hybrid renewable energy system: current status of research on configurations, control, and sizing methodologies

Piyali Ganguly , ... Aladin Zayegh , in Hybrid-Renewable Energy Systems in Microgrids, 2018

4.2.7 Computing tools

Simulation programs/ Software are the most common tools for evaluating the performance of the HRES. Using the computer simulation, the optimum sizes of the components of an HRES can be determined by comparing the performance and energy production cost for different system configurations. Several software tools are available for this purpose, such as HOMER, HYBRID2, iHOGA, HYBRIDS, RETSCREEN etc.

One of the most popular tools for optimally designing the system components of HRES is HOMER (The Hybrid Optimization Model for Electric Renewables), which is designed by National Renewable Energy Laboratory. HOMER is very frequently used for optimisation of hybrid renewable energy systems, both off grid and grid connected [101]. This software can be performing multiple analyses and can be helpful to address a wide range of design questions. It can be used to design most cost-effective systems, for optimum sizing, economic analysis. It also can be used to analyse the sensitivity of the system under changing conditions like, load variation, battery price etc. HOMER finds the combination of components with least cost to meet specific load demand. The principal tasks are simulation, optimisation and sensitivity analysis.

4.2.7.1 Simulation

HOMER simulates the operation of a system by making energy balance calculations for each of the 8,760 hours in a year. HOMER compares the electric and thermal load of every hour to the energy that the system can supply in that hour. HOMER also decides for each hour whether to charge or discharge the batteries for systems. Should the system meet the loads for the entire year, HOMER estimates the lifecycle cost of the system, accounting for the capital, replacement, operation and maintenance and interest costs.

4.2.7.2 Optimization

HOMER displays a list of feasible systems, after simulating all possible system configurations, and sort them by lifecycle cost. One can easily identify the system with the lowest cost at the top of the list. It would also be possible to scan the list for other feasible systems as per requirements.

4.2.7.3 Sensitivity Analysis

With HOMER one can study the effect of changing input conditions on system design. HOMER can perform a sensitivity analysis on almost any input by assigning more than one value to each input of interest. It would repeat the optimisation process for each value of the input so that it can be examined that how they can affect the results. One can specify as many sensitivity variables as possible, and the results can be analysed using HOMER's powerful graphing capabilities. HOMER uses hourly simulations to arrive the optimum target. It is a time-step simulator that uses hourly load and environmental data inputs for renewable energy system assessment. it facilitates the optimization of renewable energy systems based on Net Present Cost for a given set of constraints and sensitivity variables.

HOMER has been used extensively by researchers in various RES case studies [74,102–104] and in renewable energy system validation tests [69]. The operation of HOMER is simple and straightforward. However, depending upon the numbers of sensitivity variable used, the simulations can take a long time. Moreover, the program's limitation is that it does not allow the user to intuitively select the appropriate components for a system, as algorithms and calculations are not visible or accessible.

The software HYBRID2 was developed by the Renewable Energy Research Laboratory (RERL) of the University of Massachusetts. It is hybrid system simulation software, with a very precise simulation which can define time intervals from 10 min to 1 h. National Renewable Energy Laboratory recommends optimizing the system with HOMER and then, once the optimum system is obtained, improving the design using HYBRID2 [65].

iHOGA (improved Hybrid Optimization by Genetic Algorithms) is a software developed in C++ by researchers of the University of Zaragoza (Spain) for the simulation and optimization of Hybrid Stand-alone Systems of Electric Power Generation based on Renewable Energies [105]. The software can model systems with electrical energy with AC/DC load, and/or Hydrogen, as well as consumption of water from tank or reservoir previously pumped. This software is used to optimally size the system components of an energy system that includes photovoltaic generator, wind turbines, hydroelectric turbine, auxiliary generator (diesel, gasoline), inverter or inverter-charger, batteries (lead acid or lithium), charger and batteries charge controller as well as components of hydrogen (electrolyser, hydrogen tank and fuel cell). It is capable of simulating and optimizing systems of any size which is either grid-connected or standalone. The software includes the options of mono objective/ multi-objective optimization, simulation in time steps of up to 1 minute, sensitivity analysis, probability analysis, etc. Very detailed models of the components are used in this software, resulting very precise estimates of the operation of the system designed. iHOGA software includes advanced optimization methodology of genetic algorithms, which implies the possibility of obtaining the optimum system using very low computational times.

The software HYBRIDS, produced by Solaris Homes, assess the technical potential of renewable energy system for a given configuration and determines the potential renewable fraction and evaluates the economic viability based on Net Present Cost (NPC). HYBRIDS is a Microsoft Excel spreadsheet-based RES design tool that requires daily-average load and environmental data estimated for each month of the year. Unlike HOMER or iHOGA, HYBRIDS can only simulate one system configuration at a time, and it is unable to provide an optimised configuration. This software is comprehensive in terms of RES variables and the level of detail required and demands a higher level of knowledge of renewable energy system configurations than HOMER.

RETScreen is a Microsoft Excel based software tool which can evaluate the energy efficiency and the technical and financial viability of the RES projects. This tool is used to analyse the energy efficiency of the integrated system covering mainly energy production, life-cycle costs and greenhouse gas emission reduction [106].

Some more computer tools are also available for designing the components of HRES [107,108], such as The General Algebraic Modelling System (GAMS) [109], Opt Quest [110], LINDO [110,111], WDILOG2 [112], Simulation of Photovoltaic Energy Systems (Sim Pho Sys) [113], Grid-connected Renewable Hybrid Systems Optimization (GRHYSO) [114,115], and H2RES [116].

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780081024935000121

STEADY-STATE LUMPED SYSTEMS

W. Fred Ramirez , in Computational Methods in Process Simulation (Second Edition), 1997

2.2.3.3 Sparse Matrices

MATLAB has extensive sparse matrix facilities. Sparse matrices arise in many engineering problems including those of material and energy balance calculations as illustrated earlier. It is more efficient to define matrices as sparse when appropriate. If you are not sure if a matrix is sparse, then you can use the function issparse (A) which returns a value of 1 if A is sparse and 0 if it is not sparse. To define a matrix as sparse, we use the function sparse (rowpos, colpos, val, m,n)where rowpos are the positions of the nonzero row elements, colpos are the positions of the non–zero column elements, val are the values of the non–zero elements, m is the number of rows, and n the number of columns. For example, if

The result is

It is important to note that matrix operations such as ^*, +, −, \ produce sparse results if both operands are sparse. You can convert a full matrix to a sparse one by using the command

and a sparse matrix to a full one by the command

There is significant efficiency in properly using sparse matrix operations when appropriate.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780750635417500044

Energy Balances

Pauline M. Doran , in Bioprocess Engineering Principles (Second Edition), 2013

5.7 Energy Balance Worked Examples without Reaction

As illustrated in the following examples, the format described in Chapter 4 for material balances can be used as a foundation for energy balance calculations.

Example 5.4

Continuous Water Heater

Water at 25°C enters an open heating tank at a rate of 10   kg   h⁻¹. Liquid water leaves the tank at 88°C at a rate of 9   kg   h⁻¹; 1   kg   h⁻¹ water vapour is lost from the system through evaporation. At steady state, what is the rate of heat input to the system?

Solution

1.

Assemble

(i)

Select units for the problem.

kg, h, kJ, °C

(ii)

Draw the flow sheet showing all data and units.

The flow sheet is shown in Figure 5.4.

Figure 5.4. Flow sheet for continuous water heater.

(iii)

Define the system boundary by drawing on the flow sheet.

The system boundary is indicated in Figure 5.4.

2.

Analyse

(i)

State any assumptions.

–

process is operating at steady state

–

system does not leak

–

system is homogeneous

–

evaporation occurs at 88°C

–

vapour is saturated

–

shaft work is negligible

–

no heat losses

(ii)

Select and state a basis.

The calculation is based on 10   kg water entering the system, or 1 hour.

(iii)

Select a reference state.

The reference state for water is the same as that used in the steam tables: 0.01°C and 0.6112   kPa.

(iv)

Collect any extra data needed.

$\begin{array}{l}h(\text{liquid}\text{water}\text{at}88°\text{C})=368.5\text{kJ}{\text{kg}}^{-1}(\text{Table}\text{D}.1)\\ h(\text{saturated}\text{steam}\text{at}88°\text{C})=2656.9\text{kJ}{\text{kg}}^{-1}(\text{Table}\text{D}.1)\\ h(\text{liquid}\text{water}\text{at}25°\text{C})=104.8\text{kJ}{\text{kg}}^{-1}(\text{Table}\text{D}.1)\end{array}$

(v)

Determine which compounds are involved in reaction.

No reaction occurs.

(vi)

Write down the appropriate mass balance equation.

The mass balance is already complete.

(vii)

Write down the appropriate energy balance equation.

At steady state, Eq. (5.9) applies:

${\displaystyle \sum }_{\mathrm{input}\mathrm{streams}}(Mh)-{\displaystyle \sum }_{\mathrm{output}\mathrm{streams}}(Mh)-Q+W_{\mathrm{s}}=0$

3.

Calculate

Identify the terms in the energy balance equation.

For this problem W _s=0. The energy balance equation becomes:

${(Mh)}_{\mathrm{liq}\mathrm{in}}-{(Mh)}_{\mathrm{liq}\mathrm{out}}-{(Mh)}_{\mathrm{vap}\mathrm{out}}-Q=0$

Substituting the information available:

$\begin{array}{c}(10\text{kg})(104.8\text{kJ}{\text{kg}}^{-1})-(9\text{kg})(368.5\text{kJ}{\text{kg}}^{-1})-(1\text{kg})(2656.9\text{kJ}{\text{kg}}^{-1})-Q=0\\ Q=-4925.4\text{kJ}\end{array}$

Q has a negative value. Thus, according to the sign convention outlined in Section 5.2, heat must be supplied to the system from the surroundings.

4.

Finalise

Answer the specific questions asked in the problem; check the number of significant figures; state the answers clearly.

The rate of heat input is 4.93×10³  kJ   h⁻¹.

Example 5.5

Cooling in Downstream Processing

In downstream processing of gluconic acid, concentrated fermentation broth containing 20% (w/w) gluconic acid is cooled prior to crystallisation. The concentrated broth leaves an evaporator at a rate of 2000   kg   h⁻¹ and must be cooled from 90°C to 6°C. Cooling is achieved by heat exchange with 2700   kg   h⁻¹ water initially at 2°C. If the final temperature of the cooling water is 50°C, what is the rate of heat loss from the gluconic acid solution to the surroundings? Assume the heat capacity of gluconic acid is 0.35   cal   g⁻¹ °C⁻¹.

Solution

1.

Assemble

(i)

Units

kg, h, kJ, °C

(ii)

Flow sheet

The flow sheet is shown in Figure 5.5.

Figure 5.5. Flow sheet for cooling gluconic acid solution.

(iii)

System boundary

The system boundary indicated in Figure 5.5 separates the gluconic acid solution from the cooling water.

2.

Analyse

(i)

Assumptions

–

steady state

–

no leaks

–

other components of the fermentation broth can be considered water

–

no shaft work

(ii)

Basis

2000   kg feed, or 1 hour

(iii)

Reference state

H=0 for gluconic acid at 90°C

H=0 for water at its triple point

(iv)

Extra data

The heat capacity of gluconic acid is 0.35   cal   g⁻¹ °C⁻¹; we will assume this C _p remains constant over the temperature range of interest. Converting units:

$C_p(\text{gluconic}\text{acid})=\frac{0.35\mathrm{cal}}{\mathrm{g}°\mathrm{C}}\cdot \left|\frac{4.187\mathrm{J}}{1\mathrm{cal}}\right|\cdot \left|\frac{1\mathrm{kJ}}{1000\mathrm{J}}\right|\cdot \left|\frac{1000\mathrm{g}}{1\mathrm{kg}}\right|=1.47\mathrm{kJ}{\mathrm{kg}}^{-1}°{\mathrm{C}}^{-1}$

$\begin{array}{l}h(\text{liquid}\text{water}\text{at}90°\text{C})=376.9\text{kJ}{\text{kg}}^{-1}(\text{Table}\text{D}.1)\\ h(\text{liquid}\text{water}\text{at}6°\text{C})=25.2\text{kJ}{\text{kg}}^{-1}(\text{Table}\text{D}.1)\\ h(\text{liquid}\text{water}\text{at}2°\text{C})=8.4\text{kJ}{\text{kg}}^{-1}(\text{Table}\text{D}.1)\\ h(\text{liquid}\text{water}\text{at}50°\text{C})=209.3\text{kJ}{\text{kg}}^{-1}(\text{Table}\text{D}.1)\end{array}$

(v)

Compounds involved in reaction

No reaction occurs.

(vi)

Mass balance equation

The mass balance equation for total mass, gluconic acid, and water is:

$\mathrm{mass}\mathrm{in}=\mathrm{mass}\mathrm{out}$

The mass flow rates are as shown in Figure 5.5.

(vii)

Energy balance equation

${\displaystyle \sum }_{\mathrm{input}\mathrm{streams}}(Mh)-{\displaystyle \sum }_{\mathrm{output}\mathrm{streams}}(Mh)-Q+W_{\mathrm{s}}=0$

3.

Calculate

W _s=0. There are two heat flows out of the system: one to the cooling water (Q) and one representing loss to the surroundings (Q _loss). With symbols W=water and G=gluconic acid, the energy balance equation is:

$\begin{array}{c}\hfill {(Mh)}_{\mathrm{W}\mathrm{in}}+{(Mh)}_{\mathrm{G}\mathrm{in}}-{(Mh)}_{\mathrm{W}\mathrm{out}}-{(Mh)}_{\mathrm{G}\mathrm{out}}-Q_{\mathrm{loss}}-Q=0\hfill \\ \hfill {(Mh)}_{\mathrm{W}\mathrm{in}}=(1600\text{kg})(376.9\mathrm{kJ}{\mathrm{kg}}^{-1})=6.03\times {10}^5\mathrm{kJ}\hfill \\ \hfill {(Mh)}_{\mathrm{G}\mathrm{in}}=0(\text{reference}\text{state})\hfill \\ \hfill {(Mh)}_{\mathrm{W}\mathrm{out}}=(1600\text{kg})(1.47\mathrm{kJ}{\mathrm{kg}}^{-1})=4.03\times {10}^4\mathrm{kJ}\hfill \end{array}$

(Mh)_Gout at 6°C is calculated as a sensible heat change from 90°C using Eq. (5.12):

$\begin{array}{c}\hfill {(Mh)}_{\mathrm{G}\mathrm{out}}=M{C}_p(T_2-T_1)=(400\text{kg})(1.47\mathrm{kJ}{\mathrm{kg}}^{-1}°{\mathrm{C}}^{-1})(6-90)°\mathrm{C}\hfill \\ \hfill \therefore {(Mh)}_{\mathrm{G}\mathrm{out}}=-4.94\times {10}^4\mathrm{kJ}\hfill \end{array}$

The heat removed to the cooling water, Q, is equal to the enthalpy change of the cooling water between 2°C and 50°C:

$Q=(2700\text{kg})(209.3-8.4)\mathrm{kJ}{\mathrm{kg}}^{-1}=5.42\times {10}^5\mathrm{kJ}$

These results can now be substituted into the energy balance equation:

$\begin{array}{c}\hfill (6.03\times {10}^5\text{kJ})+(0\text{kJ})-(4.03\times {10}^4\text{kJ})-(-4.94\times {10}^4\text{kJ})-Q_{\mathrm{loss}}-5.42\times {10}^5\mathrm{kJ}=0\hfill \\ \hfill \therefore Q_{\mathrm{loss}}=7.01\times {10}^4\mathrm{kJ}\hfill \end{array}$

4.

Finalise

The rate of heat loss to the surroundings is 7.0×10⁴  kJ   h⁻¹.

It is important to recognise that the final answers to energy balance problems do not depend on the choice of reference states for the components. Although values of h depend on the reference states, as discussed in Section 5.3.1 this dependence disappears when the energy balance equation is applied and the difference between the input and output enthalpies is determined. To prove this point, any of the examples in this chapter can be repeated using different reference conditions to obtain the same final answers.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780122208515000058

Combustion process calculation for fuels, biomass, wastes, biosludge, and biocarbons

Yen-Hsiung Kiang , in Fuel Property Estimation and Combustion Process Characterization, 2018

7.11 End Notes

Theoretical bases and implementation methodologies for combustion system calculations are presented in this chapter. Examples are used to illustrate these calculation methods. Models for the required data for energy balance calculation, such as enthalpy of formation of combustion products, are also included.

In order to carry out material and energy balance calculation, besides the heating value and composition of the fuels, the types of the products of complete combustion must be known. In this chapter, the primary products of complete combustion for the reacting elements are presented. Since the concentrations of the incomplete combustion products are significantly smaller, compared with the products of complete combustion, they are normally ignored.

The material balance calculation for combustion processes can be carried out traditionally through balancing chemical reaction equations. However, this chapter presented a tabular format of material balance calculation. This tabular format calculation methodology can be programmed into Excel format for computerized calculations.

For energy balance, one of the major objectives is to calculate the combustion temperature. In this chapter, three methodologies are presented for the calculation of combustion temperature. One of the three methodologies is the direct calculation of combustion temperature by the results of material balance and the relevant enthalpy data of combustion product gases. A simplified calculation is also presented that can be used to calculate the combustion temperature directly. The third method is to use the iterative method.

The integration of combustion process characterization methodologies, including sample of basic process flow diagram and material and energy balance table is also presented in this chapter. These two documents form the basis for combustion process design and operation.

When carrying out the overall material and energy balance calculation for the combustion processes, sometimes it is necessary to determine the auxiliary fuel requirement (or heat removal requirement) for the combustion processes to maintain the preassigned combustion temperature. A total of five models that can be used to carry out overall material and energy balance calculations are presented in this chapter.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128134733000076

Hardmetals

Alex V. Shatov , ... S.A. Firstov , in Comprehensive Hard Materials, 2014

1.10.3.4.3 Sigl, Exner and Fischmeister (SEF) Comprehensive Model of Energy Balance of All the Fracture Modes

The SEF semiempirical model (Sigl, Exner, & Fischmeister, 1986; Sigl & Exner, 1987; Sigl & Fischmeister, 1988 ) is the most comprehensive energy balance calculation of all the fracture modes for WC-Co cemented carbides. The SEF model is based on the Sigl and Exner empirical correlations of the fracture surface area fractions $A_{\text{A}}^{\text{i}}$ for the fracture modes (2)-(6) and it does take into account the fracture energy of the WC phase w _C and w _C/C estimated at about 50   J   m⁻² by Eqn (26) into the energy balance (25).

(38) $G_{1\text{C}}^{\text{SEF}}=\left({\overline{\text{r}}}_{\text{B}}·A_{\text{A}}^{\text{B}}+{\overline{\text{r}}}_{\text{B}/\text{C}}·A_{\text{A}}^{\text{B}/\text{C}}\right)·{\overline{\sigma }}_{\text{f}}^{\text{B}}+\left(A_{\text{A}}^{\text{C}/\text{C}}+A_{\text{A}}^{\text{C}}\right)·w_{\text{C}}$

where ${\overline{\text{r}}}_{\text{B}}$ and ${\overline{\text{r}}}_{\text{B}/\text{C}}$ are the mean half-size of the plastic zone of the fractured binder ligaments at the transgranular fracture of the binder phase B and along the carbide-binder interfaces B/C, respectively, whereas ${\overline{\sigma }}_{\text{f}}^{\text{B}}$ is the mean flow stress of the binder ligaments from plastic deformation and up to the formation and fracture of dimples.

The SEF model calculates the mean flow stress of the binder ligaments ${\overline{\sigma }}_{\text{f}}^{\text{B}}$ as a sum of the yield ${\sigma }_{\text{y}}^{\text{B}}$ and the strain hardening ${\sigma }_{\text{sh}}^{\text{B}}$ stresses of the binder ligaments.

(39) ${\overline{\sigma }}_{\text{f}}^{\text{B}}={\sigma }_{\text{y}}^{\text{B}}+{\sigma }_{\text{sh}}^{\text{B}}$

The yield stress is assumed to follow the Hall-Petch dependence (Hall, 1952; Petch, 1953) with the friction stress ${\sigma }_0^{\text{B}}$ and the Hall-Petch strengthening factor $K_{\text{B}}^{\text{HP}}$ for the binder phase.

(40) ${\sigma }_{\text{y}}^{\text{B}}={\sigma }_0^{\text{B}}+K_{\text{B}}^{\text{HP}}·{\lambda }^{-1/2}$

The SEF model estimates the value of the friction stress at about ${\sigma }_0^{\text{B}}=480\text{MPa}$ from the yield stress of Co-W-C alloy with the solid solution of W and C matching the solution in the binder phase that is determined by the magnetic saturation of WC-Co alloys (Roebuck, Almond, & Cottenden, 1984). The strengthening factor of the binder phase $K_{\text{B}}^{\text{HP}}=1.35\text{MPa}·{\text{m}}^{1/2}$ is calculated from the value of the Hall-Petch hardening factor of the binder phase $H_{\text{Binder}}^{\text{HP}}=3.94\text{MPa}·{\text{m}}^{1/2}$ in the LG model for hardness (Gurland, 1979; Lee & Gurland, 1978) by using the conventional relation between the yield stress and the hardness H.

(41) ${\sigma }_{\text{y}}\approx H/3$

Substitution of ${\sigma }_0^{\text{B}}$ and $K_{\text{B}}^{\text{HP}}$ into Eqn (40) gives the yield stress ${\sigma }_{\text{y}}^{\text{B}}$ values between 2.2 and 3.7   GPa depending on the mlibp λ that decreases from 0.74 to 0.24   μm (Sigl, Exner, & Fischmeister, 1986; Sigl & Fischmeister, 1988).

The SEF model estimates the strain hardening of the Co-binder ligaments ${\sigma }_{\text{sh}}^{\text{B}}$ from assumption that the dimples have parabolic depth profile and that the fracture occurs when the stress reaches the saturation level, at which the dislocation has to slip through the saturated number of stacking faults in the binder ligament.

(42) ${\sigma }_{\text{sh}}^{\text{B}}\approx 4.7\text{GPa}$

Substitution of all the estimated values into Eqn (39) gives the mean flow stress of the binder ligaments ${\overline{\sigma }}_{\text{f}}^{\text{B}}$ in the range from 7.0 to 8.4   GPa for the mlibp λ that decreases from 0.74 to 0.24   μm (Sigl, Exner, & Fischmeister, 1986; Sigl & Fischmeister, 1988).

The value of the mean plastic zone half-size for B/C fracture ${\overline{\text{r}}}_{\text{B}/\text{C}}$ is empirically measured as a half depth of the shallow binder ligament dimple along the carbide-binder interface at about 0.05   μm. Calculation of the mean plastic zone half-size for the fracture through the binder phase ${\overline{\text{r}}}_{\text{B}}$ requires solving integral equations and detailed measurement of the distribution of the linear intercepts of the binder ligaments. The mean plastic zone size ${2·\overline{\text{r}}}_{\text{B}}$ calculated by the SEF model never exceeds the mlibp λ and ranges between 0.236 and 0.593   μm for the mlibp λ between 0.24 and 0.74   μm. The value of ${2·\overline{\text{r}}}_{\text{B}}$ is practically equal to λ at lower λ  <   0.3   μm, but ${2·\overline{\text{r}}}_{\text{B}}$ becomes visibly below λ with further increase of λ. Unfortunately, no empirical approximation for the dependence of ${2·\overline{\text{r}}}_{\text{B}}$ on λ is provided by the SEF model.

The SEF model covers most of the processes of the fracture and provides viable explanation of the fracture toughness of WC-Co cemented carbides. Nevertheless, despite its comprehensiveness and thorough analysis of the deformation and crack propagation, the SEF model does not allow calculating the fracture toughness from the stereological parameters of the microstructure, but requires tedious empirical measurements of the mean half-sizes ${\overline{\text{r}}}_{\text{B}}$ and ${\overline{\text{r}}}_{\text{B}/\text{C}}$ of plastic zones for B and B/C fracture modes, respectively, in addition to the empirical measurement of the fracture surface area fractions $A_{\text{A}}^{\text{i}}$ by Eqns (2)-(6). Calculation of the approximated dependences of the plastic zone sizes could facilitate further analysis and extension of the SEF model to other materials.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080965277000106

Modeling and analysis of an islanded hybrid microgrid for remote off-grid communities

Shubham Tiwari , ... Jai Govind Singh , in Residential Microgrids and Rural Electrifications, 2022

10.4 Economic analysis through HOMER

HOMER takes many inputs, such as monthly or early load data, solar data, or wind data, to simulate different combinations of generation. It produces comparative results by formulating energy balance calculations for every hour of the year (i.e., 8760  hours). For every hour it calculates and compares electrical and thermal demands on the system and subsequently computes the optimal flow of power from each resource.

In this study, our focus was to calculate and compare the COE of the system in different scenarios. The scenarios considered here are as follows. First, only a diesel generator was considered as the source with no renewables. Second, solar PV generation was added in the system. In third scenario a hybrid diesel-solar PV-battery storage system was considered. Analysis from HOMER focuses on the effects on the COE of adding external additional battery storage.

HOMER defines COE as the average cost per kilowatt-hour of useful electrical energy produced by the system. To calculate the COE, HOMER divides the annualized cost of producing electricity (the total annualized cost minus the cost of serving the thermal load) by the total electric load served, using Eq. (10.2):

(10.2) $\mathrm{COE}=\frac{{\mathrm{COE}}_{\mathrm{ann},\mathrm{tot}}-c_{\mathrm{boiler}}-H_{\mathrm{served}}}{E_{\mathrm{served}}}$

where

COE_{ann, tot} is total annualized cost of the system ($/year)

C _boiler is the boiler marginal cost ($/kWh)

H _served is the total thermal load served (kWh/year)

E _served is the total electrical load served (kWh/year)

To analyze the system COE, the load profile and solar data shown Figs. 10.5 and 10.6, respectively, were fed to the software. Additional information from Bhatt and coworkers regarding initial capital cost and per unit cost of resources were considered (Bhatt et al., 2016). Apart from COE, other parameters such as diesel output consumption, total renewable energy output, are also discussed in Section 10.5.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780323901772000049

Combustion process characterization for fuels, biomass, wastes, biosludge, and biocarbons

Yen-Hsiung Kiang , in Fuel Property Estimation and Combustion Process Characterization, 2018

8.1 Introduction

In Chapter 7 , Combustion process calculation for fuels, biomass, wastes, biosludge, and biocarbons, the material and energy balance methods for combustion processes are presented. The results of material and energy balance calculations can be used for combustion system and equipment sizing purposes. However, these material and energy balance calculation results cannot provide complete combustion process characterization information. To complete the combustion process characterization process, fates of other products of complete and incomplete combustion have to be identified. These other properties of products of combustion are used for the selection of equipment configuration, the material of construction selection, heat recovery systems, and air pollution control systems selection and design.

In this chapter, the fates of halides contained in the fuels are estimated through chemical equipment equilibrium calculations. Chemical reaction equilibrium constants estimation models are provided to determine the concentrations of hydrogen fluoride, hydrogen chloride, hydrogen bromide, and hydrogen iodide as well as their corresponding concentrations of fluorine, chlorine, bromine, and iodine.

Although the majority of sulfur compounds presented in the products of combustion is in the form of sulfur dioxide, some quantities of sulfur trioxide do exist. A model is presented in this chapter to estimate the chemical reaction equilibrium concentrations of sulfur trioxide in the combustion product gases.

Nitrogen oxides are major pollutants emitted from the combustion processes. Models to estimate the equilibrium concentrations of nitrogen oxides (nitrogen monoxide and nitrogen dioxide) are presented in this chapter. Also, reaction kinetic models to estimate the concentrations of prompt nitrogen oxides, thermal fixation nitrogen oxides, and fuel nitrogen produced nitrogen oxides are included. An empirical model is presented in this chapter which can be used to estimate the formation of nitrogen oxides from fuel nitrogen. The model to estimate the equilibrium distribution between nitrogen oxide and nitrogen dioxide are also presented in this chapter.

Products of incomplete combustion and dioxins are toxic combustion products. Their formation mechanisms are reviewed and discussed in this chapter. This information can be used for the reduction and control of products of incomplete combustion and dioxins. The test results of the concentrations of dioxins in the combustion processes are reported in this chapter. The dioxin test results include the release of dioxin quantities from both old and modern incineration systems.

Dew points are important factors in the prevention of the material of construction from corrosion. Acid gases dew points in the product of combustion can be estimated with models presented in this chapter.

The fates of metals contained in the fuels are discussed in this chapter.

Molten slags are always a problem in the design of the combustion processes. The understanding of the melting temperatures of these molten slags provides valuable information for the combustion process design, including the selection of the combustion temperatures and equipment configurations. The estimations of eutectic points of the metal compound mixtures are presented.

In the chemical industry, there is a special group of waste liquids that contain alkaline metals, such as sodium. The incineration of this group of waste liquids (i.e., from caprolactam plants or acrylonitrile plants) presents a unique problem that requires special design considerations. This chapter discusses the incineration system design and heat recovery systems of this waste group.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128134733000088

Review of System Design and Sizing Tools

Santiago Silvestre , in McEvoy's Handbook of Photovoltaics (Third Edition), 2018

4.1 Sizing tools

Sizing tools allow PV systems to be dimensioned taking into account energy requirements, site location, and system costs (Table 3 ). Most of these software tools are relatively simple and help users automatically solve energy balance calculations considering different combination of PV system components, including batteries, modules, and loads. These tools are usually implemented using spreadsheets at different level of complexity and offer a first approach for the evaluation of specific PV system applications. In their basic form these tools are easy to use, and some solar cell manufacturers and vendors of PV system components offer this kind of tool to potential customers (often via the Internet) to enable customers to adapt PV system components to their products. Similar sizing tools are also available that focus on specific studies for complete sets of modules, batteries, inverters, electronic power conditioning components, and loads from the principal manufacturers in the PV market.

Table 3. System-sizing programs

Program	Source	Description
MaxDesign (Version 3.5.2)	SolarMax [31]	MaxDesign is an interactive photovoltaic (PV) system design tool for all SolarMax inverters. It offers design variants, finding out which configurations are the best for a specific PV system. Results include dimensioning, yield calculation, and analysis of DC line losses
NSol (Version Vx)	NSol [32]	It includes modules for stand-alone PV, PV–generator hybrids, and grid-tied PV. The stand-alone version includes the loss-of-load (LOL) probability statistical analysis
PVGIS	European Commission [33]	Web application to estimate the performance of PV systems located in Europe
PVWATTS (Version 5.3.8)	NREL [34]	An Internet-accessible tool that calculates the electrical energy produced by a grid-connected PV system for locations within the United States and its territories
RETScreen (Version 4)	CANMET Energy Diversification Research Laboratory [35]	RETScreen International is a renewable energy decision-support and capacity-building tool. Each RETScreen renewable-energy-technology model, including hybrid PV systems, is developed within an individual Microsoft Excel spreadsheet workbook file. The workbook file is in turn composed of a series of worksheets. These worksheets have a common format and follow a standard approach for all RETScreen models. In addition to the software, the tool includes product, weather, and cost databases; an online manual; a Web site; project case studies; and a training course
SAM (Version 2016.3.14)	NREL [36]	SAM makes performance predictions for grid-connected solar, small wind, and geothermal power systems and economic estimates for distributed energy and central generation projects
Solar configurator (Version 1.0.1635.1)	Fronius [37]	Allows the analysis of the different configuration possibilities: PV module types, strings, tilt angles, and wirings
Solar Pro (Version 4.5)	Laplace System Co., Kyoto, Japan [38]	Program enables users to gauge the effects of shade from adjacent buildings or objects so as to determine the optimum positioning and scale of the PV modules and system. It also allows for estimating the I–V curve of solar cell modules based on the electrical characteristics of various module designs, and it calculates the electrical yield of systems based on geographic location and atmospheric conditions over a wide array of possibilities
Sunny Design	SMA [39]	Sunny Design allows definition of the PV generator, definition of system location, module type, orientation, and power generated or number of PV module and the determination of the typical investor (SMA). Results help determine the characteristics of performance of a year of service

NREL, National Renewable Energy Laboratory, USA; SMA, SMA Solar Technology AG.

More sophisticated sizing tools are also available in the market, some offering the possibility of optimizing the size of each PV system component. More detailed analysis can then be carried out of the energy flows in the PV system and the determination of critical periods along the year. Of particular interest is often the deficit of energy associated with these periods of time and minimizing the final cost of the system for a specific application.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128099216000185

lambertconumpen.blogspot.com

Source: https://www.sciencedirect.com/topics/engineering/energy-balance-calculation

Share this post