CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 70, 2018 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Timothy G. Walmsley, Petar S. Varbanov, Rongxin Su, Jiří J. Klemeš 
Copyright © 2018, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-67-9; ISSN 2283-9216 

Comparative Study on Heat Generation During Compressible 

Airflow Through Heatproof Nozzles 

Krzysztof J. Wołosz*, Jacek Wernik 

Warsaw University of Technology, Faculty of Civil Engineering, Mechanics and Petrochemistry, Łukasiewicza 17, 09-400 

Płock, Poland 

Krzysztof.Wolosz@pw.edu.pl 

The study considers industrially applicable nozzles which are used as an auxiliary equipment in pneumatic 

pulsator systems for unclogging outlets of silos for loose materials. The aim is to determine the amount of heat 

(energy loss) which is generated during one work cycle of the system. Presented study compares energy loss 

during compressible (max. M = 1.7) and transient airflow through two types of nozzles: with and without a baffle. 

Investigation in this field has not been carried out so far and present-day designing process is significantly based 

on heuristic knowledge. The energy loss is calculated by using results of numerical simulation. The OpenFOAM 

CFD toolbox with the finite volume method has been used for that purpose with thermodynamically ideal gas 

model. The energy analysis and heat generation calculation have been based on the first principles of 

thermodynamics. For initial and boundary condition, authors’ previous work, which considered rapid airflow from 

a vessel to the atmosphere, has been partially utilized. Instantaneous results of simulation indicate the high 

energy rates but they are available for a very short time frame. The values of the total heat generated during 

gas conversion within the nozzle show that both nozzle produces a small amount of energy loss which would 

cause a slight warm-up of them. 

1. Introduction 

Investigations of a flow of compressible medium through a nozzle usually leads to a determination of flow 

parameters and actually possibility of their intensification. Energy analysis of gas flow and possibility of its 

utilization in the nozzles is not commonly carried out. The most frequent analyses of energy efficiency concern 

systems of equipment which are mostly used in industry. In this article, the authors compare the amount of 

energy lost in the nozzle of a pneumatic pulsator system. In systems for clearing silo outlets, compressed air is 

used for to destroy the unfavourable structure in loose material bed such as archings, bridges and others. The 

air, which is pressurized in the pressure accumulator, is rapidly released and a pneumatic impact is produced. 

This impact is a force that counteracts cohesion forces which are specific for loose and bulk materials. Systems 

for clearing silo outlets are an important element of safety and permeability of transport systems in industrial 

plants. Visualization of a silo with pneumatic pulsator system placed on its walls can be observed in Figure 1. 

To make the investigated object easier to understand, the nozzle is zoomed. 

In pneumatic pulsator systems, nozzles are used to direct the airflow and often to protect the pulsator itself 

against the heat generated inside the silo. This is the case, for example, in cement plants in silos which contains 

clinker from the lime kiln. Different types of nozzles are used which differing in size, shape and function. The 

article presents a comparative study of two straight nozzles. The difference between them is a presence of 

reinforcing baffles placed along air stream in one of them. 

In the nozzles, due to the principal of operation of the industrial pulsator, there is a compressible, unsteady 

airflow. Accompanying phenomena are related to thermodynamic gas conversions, which in their description 

must take into account the formation of shockwaves. The article by Honma et al. (2003) is of high value in this 

field due to the methods used and the comparison of experimental results with numerical ones. The authors 

considered unsteady outflow of gas to the atmosphere. Ota et al. (2012), and the related article by Inage et al. 

(2013), presented also very similar works in which the authors primarily focused on measurement methods and 

imaging capabilities of the shockwave and related phenomena. 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1870026 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Wołosz K.J., Wernik J., 2018, Comparative study on heat generation during compressible airflow through heatproof 
nozzles , Chemical Engineering Transactions, 70, 151-156  DOI:10.3303/CET1870026   

151



 

Figure 1: Pulsator system visualization on the background of a silo with the nozzle zoomed. 

Investigations which consider nozzles and compressible flow mainly focus on the studies of Laval-type 

convergent-divergent nozzles. Due to the wide range of its applications, these studies most often focus on the 

effects of changes in geometry on the flow parameters and the studies consider specific industrial applications 

such as, for example, cooling systems (Lin et al., 2013), solar air conditioning systems (Hemidi et al., 2009) as 

well as improving the overall effectiveness of ejectors (Zhu et al., 2009). 

Applications of research methods related to compressible flows in nozzles for heavy industry have been initiated 

by Wołosz and Wernik (2014) and then developed in a work focusing on the methodical aspects (Wołosz and 

Wernik, 2016b). The results of other studies related to pneumatic pulsators presented by Wołosz and Wernik 

(2016a) have also been used in this work, whereas the methods given by Wołosz (2018) to through-flow systems 

(with inlet and outlet) are developed. 

The application of energy calculations is a novelty in systems in which the unsteady compressible gas flow is 

present. Literature review shows that researchers most often focus on nozzle parameters. However, they do 

not take into account the energy efficiency and its impact on a system in which these nozzles are located. The 

present study is an extension of the research to include these aspects. The article shows the magnitude of the 

energy losses associated with compressible transient airflow through a closed channel. It should also be 

emphasized that the energy efficiency is very rarely considered in this area of industrial issues. 

2. Numerical Model 

For the energy analysis, the geometric models of nozzles have been used. In both cases, the computational 

domain covers the inside of the nozzle with its symmetry plane taken into account. Each nozzle is mounted to 

a pulsator with a nominal diameter of 150 mm and the main dimensions are: length along the air stream 320 mm, 

width – 487 mm, and height – 325 mm. Figure 2 shows the visualization of nozzles cross-sections with the 

computational mesh applied. In both cases, the meshes exceeded 1,200,000 cells. Mesh generation as well as 

numerical calculation have been proceeded by using OpenFOAM. The OpenFOAM is an open source toolbox 

for Computational Fluid Dynamics (CFD) with finite volume method applied for solving governing equations of 

fluid flow. 

Numerical simulation has been carried out by using formulas that are mathematical expression of basic 

conservation laws, as follows: 

• mass conservation (continuity equation): 

, 
(1) 

• momentum conservation law: 

, 
(2) 

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• energy conservation law: 

, 
(3) 

where ρ is density, u – velocity vector, μ – dynamic viscosity, p - pressure, t – time, e – internal energy of the 

gas, and λ denotes thermal conductivity. 

 

a) 

 

b) 

 

Figure 2: Nozzles with computational mesh applied: a) plain (without a baffle), b) with a baffle. 

Eq(1) - Eq(3) are properly solved only for laminar flows. Air velocity reaches high values and the maximal Mach 

number is M = 1.7 in both investigated cases. Therefore, it is necessary to apply a model of turbulence to gain 

reliable results. For that purpose, standard k-ε model has been used (Launder and Jones, 1972) which is well 

validated in engineering aplications. 

Air is a compressible medium which is commonly modelled as a gas which obey ideal gas equation: 

, 
(4) 

where R – gas constant, cp – specific heat at constant pressure, cv - the one at constant volume, and T denotes 

temperature. 

Internal energy of the gas is calculated by using the following expression: 

, (5) 

and the entire aforementioned system of equations must be complemented by equation of heat conduction, 

which is Fourier’s law: 

, (6) 

where q is heat flux density. 

The system of equations Eq(1) - Eq(6) was numerically solved by using finite volume method. The equations 

were discretised by using an appropriate discretisation scheme. In this study for spatial discretisation, Gauss 

linear scheme was used, which is second-order accurate. Because the study considers transient airflow, 

precise, second-order accurate, Crank-Nicolson scheme for time discretisation was utilized. 

Properly stated numerical problem can only be solved when boundary and initial conditions are applied. In the 

present study, the driving force of the flow is pressure difference on the both sides of the nozzles. The inlet 

pressure values come from the authors’ previous study on emptying of the pressure accumulator (Wołosz and 

Wernik, 2016a). These results made it possible to calculate inlet values of pressure by using formula for 

calculating stagnation and static pressure ratio according to theory of shockwaves as follows: 

 

(7) 

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where pp denotes patch (boundary) pressure, p0 – stagnation pressure, and γ is adiabatic index. The value of 

pressure has a crucial influence on the value of energy of the gas at the inlet of the nozzle. 

Other necessary boundary conditions were as follows: 

• initial inlet and outlet temperature is 300 K (Dirichlet boundary condition), 

• inlet pressure and velocity gradients equal zero (Neumann boundary condition), 

• walls of the nozzle are assumed adiabatic and impermeable. 

It is very important to proceed numerical model verification and solver validation during numerical simulations. 

For the solver validation, the benchmark model has been built in OpenFOAM on the basis of experimental setup 

reported by Giglmaier et al. (2013) and the results have been compared. The error estimation has been carried 

out by using procedure proposed by Freitas (2002). 

3. Energy and Energy Loss 

The amount of energy which is delivered to and received from the nozzle can be determined by using the First 

Law of Thermodynamics (FLT). For the purpose of this study, it is convenient to express this Law by its integral 

form, as follows: 

 
(8) 

The first integral in Eq(8) is of no value if there are neither chemical nor nuclear reaction, i.e. there are no internal 

heat sources. Airflow through the nozzle proceeds very fast, therefore, adiabatic gas conversion can be 

considered. With these assumptions taken into account, FLT looks as follows: 

 
(9) 

Because of impermeability of the nozzle walls, the boundary area A is limited to the inlet and the outlet of the 

nozzle. The values of density, temperature, pressure and velocity are changing along the considered area. 

Therefore, to predict total energy flux, the values have been area-weighted averaged, and the FLT is: 

 
(10) 

where un is average velocity normal to the boundary area A. 

4. Results and Discussion 

The most important flow parameter regarding speed of the entire process is pressure, both inlet and outlet, and 

their changes in time. Averaged values of pressure for both types of investigated nozzle and their time 

relationship are shown in Figure 3. Furthermore, time dependence of stagnation pressure p0, which for both 

types of nozzle has been the same, is shown in Figure 3a. It can be noticed that there is a difference between 

inlet pressure values after simulation time of 0.4 ms. From that time, inlet pressure is lower in the nozzle with 

the baffle then in the one without it. Inlet pressure is calculated by using Eq(7), so the baffle causes velocity 

increase by narrowing flow cross-section. This is the case where flow parameters influence the inlet boundary 

conditions. That is why it is very important to state the initial and boundary conditions properly. 

 
(a) 

 
(b) 

Figure 3: Pressure time distribution at: a) inlet, b) outlet of the nozzles. 

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The energy analysis is subsequently carried out based on the results of numerical simulation. The values of 

energy fluxes at inlets and outlets of each nozzle are obtained in this analysis. The plots of these fluxes and 

their time relationship can be seen in Figure 4. It should be noticed that the scale of the horizontal axis differs 

slightly for the inlet and outlet plots. Because the time needed to overcome the distance of the nozzle length is 

about 0.5 ms, the outlet values equal initial ones up to this time. Therefore, due to the increased readability of 

the plots, the scale of the horizontal axis is shortened in Figure 3 and 4. 

 

 
(a) 

 
(b) 

Figure 4: Time distribution of energy flux at inlet and outlet of the nozzle. 

Inlet energy fluxes for both nozzle types start to differ since 0.4 ms of simulation time. It follows directly from the 

difference of inlet pressure what has been mentioned before. Therefore, it might be claimed that the presence 

of the baffle generates inlet gas energy decrease by causing inlet pressure decrease. 

The highest recorded value of the inlet energy flux is above 3 MW. This value is more than twice less at the 

outlet, i.e. approx. 1.5 MW. The energy fluxes shown in the plots of Figure 4 do not give the answer what is the 

total energy of the process for both nozzles. Due to slightly different energy flux characteristics, it is difficult to 

directly estimate the energy utilization and compare it between both types of nozzle. Furthermore, the highest 

energy flux is available at very short time frame. For example, 3 MW are available only for period 0.001 ms. For 

that reason, the total energy must be determined for the whole process from 0 to 500 ms, i.e. the time after 

which the flow in the nozzles stops. Simulation time could be shorter, however, to restrict the calculation error 

and due to slow coincidence of temperature results, it was decided to extend simulation time to 500 ms. 

Time derivative of energy gives energy flux, thus, to calculate total energy at inlets and outlets, the energy fluxes 

need to be integrated in time. Since numerical simulation gives discretised results, the integral can be converted 

into sum as follows: 

 

(11) 

After this operation, the values of total energy can be determined for inlets and outlets of both types of nozzle 

and these values can compared in order to determine the energy loss according to the following relation: 

 (12) 

The results of total energy at inlets and outlets and heat generated (energy loss) are presented in Table 1 below: 

Table 1: Results of energy efficiency analysis 

Nozzle type Inlet energy [J] Outlet energy [J] Energy loss [J] Energy loss [%] 

w/ baffle 10,399 10,322 76.9 0.74 

w/o baffle 14,059 13,828 230.7 1.64 

 

There is a noticeable lower inlet energy for the nozzle with the baffle which is obvious since the presence of the 

baffle itself. However, the lower value of losses for the nozzle with the baffle is unexpected. The presence of 

the baffle should suggest an increase in losses. Percentage comparison shows that the nozzle without the baffle 

generates more energy loss. This might suggest that there is a chocked flow inside the nozzle, which generates 

additional losses. 

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5. Conclusions 

During this study, a method has been developed for evaluating how much energy, which is carried with 

compressed air, is lost during the unsteady, compressible flow through the heatproof nozzles of an industrial 

pneumatic pulsator system. The amount of these losses, which must be converted into heat eventually, is 

regarded as small. Depending on the nozzle type, the amount of heat generated in the nozzle does not exceed 

1 % for the nozzle with the baffle and 2 % for the nozzle without the baffle. These values are not large and 

should not affect on the quality of the nozzle functioning. What may affect the quality of their functioning are the 

conclusions resulting from the comparison of the values obtained for individual nozzles. In spite of expectations, 

the energy loss is greater when using the nozzle without the baffle. This will lead to further research on the use 

of other nozzle shapes. However, a simple conversion towards the de Laval nozzle cannot take place due to 

the unsteady nature of the pneumatic impact phenomenon, which is encountered in industrial pneumatic 

pulsators. Therefore, in future works, it is also expected to perform flow simulations established for both 

considered nozzles using the initial boundary conditions of this study and to compare the energy values for 

steady and unsteady flows. 

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