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HEAT TRANSFER MAPPING OF DRUM
FLAKER(CHILLROLLS)
A MAJOR PROJECT REPORT
Submitted in partial fulfillment of the
requiremen...
2
CANDIDATE’S DECLARATION
I hereby declare that the work, which is being presented in this dissertation, Entitled “TOPIC’’...
3
ACKNOWLEDGEMENT
I am using this opportunity to express my gratitude to everyone who supported me
throughout the course o...
4
ABSTRACT
This report explores the parametric influence of spray cooling for thick walled metal drum with a
thin layer of...
5
Contents
CANDIDATE’S DECLARATION.........................................................................2
ACKNOWLEDGEME...
6
2.8 Applications...................................................................................................22
2....
7
LIST OF FIGURES
No of Figure Name of Figure Page No
2.1 Drum flaker (Chill Roll) 15
2.2 Over of Drum flaker (Chill Roll)...
8
LIST OF TABLES
No of Table Name of Table Page No
2.5.1 Table of Parameter and
Tendency
18
9
NOMENCLATURE
Vz(z) mean z- velocity of liquid inside pipe
Δ(z) thickness of the (“boundary” layer )of liquid in the drum...
10
Greek symbols
λw (Tcake) latent heat of vaporization of water at cake temperature r = (Rp +tm+tcake)
ρ density of water
11
1. INTRODUCTION
Hindustan Unilever Limited (HUL) is India's largest Fast Moving Consumer Goods Company
with a heritage ...
12
Lipton's links with India were forged in 1898. Unilever acquired Lipton in 1972, and in 1977
Lipton Tea (India) Limited...
13
1.1 Motivation for the work
In Hindustan unilever Limited, in NMB plant (soap plant). The drum flaker is used primarily...
14
2. LITERATURE REVIEW
The drum flaker is used primarily to process chemical and pharmaceutical products.
However, more a...
15
Product is then scraped off the rolls, by the doctor blades, in form of flakes and fall onto a
discharge conveyor belt....
16
Figure 2.2 Overview of Drum Flaker (Chill rolls)
2.2 Features of the Drum Flaker (Chill rolls)
The Drum Flaker enables ...
17
impossible in normal operation. The drum design ensures equal heat transfer over the entire
drum surface, ensuring unif...
18
2.4 The Purpose of Flaking
The purpose of flaking is to cool and solidify fusion liquid. The solidified product is shap...
19
Fig. 2.3 Spray nozzle configuration
This is a spray nozzle arrangement along the axis of a cylindrical thick-walled tub...
20
A wide variety of spray nozzle applications use a number of spray characteristics to describe the
spray.
2.6.1 Hollow c...
21
.
Figure 2.7.1: Full water type
2- Cooling water flows in the jacket. Without being left, lukewarm water will be discha...
22
2.8 Applications
Chilled roll flaker is a multi-purpose drying device. It is generally used in industries where the
pro...
23
3. MATHEMATICAL MODELLING
3.1 Methodology
The objective of my project is to study and Construct a heat transfer model o...
24
Learning about mathematical modelling is an important step from a theoretical mathematical
training to an application-o...
25
Integrate above equation over z
Vzδ = qwz + C1
Boundary conditions
At z = 0, Vz = δ = 0 therefore C1 = 0
We get
Vzδ = q...
26
qw. Twf =
d
dz
(qwz. T) +
Q(z)
Cpwρw
Or
qw. Twf = qw
d
dz
(z. T) +
Q(z)
Cpwρw
Or
qw. Twf = qw [z
dT
dz
+ T. 1] +
Q(z)
C...
27
Fig 3.6.1 Assumed Flow Model Inside The Drum (Heat and Fluid)
3.7 Mass Balance For The Cake
Assuming only radial direct...
28
Mass Balance (steady state)
In-Out=0
-
-Dab.2πrdz.
dya
∗
dr
-[- Dab.2πrdz.
dya
∗
dr
- Dab.2πdz.
d
dr
(r .
dya
∗
dr
)dr ...
29
Or
d
dr
[r
dTcake
dr
] = 0 (3.7)
Boundary conditions
At r = Rp + tm Tcake = T0
At r = Rp + tm + tcake
−Kcake. 2π(Rp + t...
30
4. SOLUTION
4.1 Final Equations To Solve
Vzδ = qwz (3.2)
qw (Twf − T) = qw z
dT
dZ
+
Q(z)
Cpwρw
(3.4)
Q(z) =
Km (T(z) −...
31
Boundary Conditions
At r = Rp + tm
Tcake = T0
At r = Rp + tm + tcake
hair(Tair − Tcake) − Kcake
dTcake
dr
= λw(Tcake). ...
32
At r = r1 Tcake = T0
At r = r2 −k
dTcake
dr
= λw. Tcake. ky(ya
∗∗
− yab) − hair(Tair − Tcake)
C1 = −
r2
k
[λw. Tcake. k...
33
5. RESULT AND DISCUSSION
This study examined the problem of spray cooling thick walled metal. An analytical model was
c...
34
Figure 3.2 Shows the change in temperature (T) profile (mean temperature of the water flowing
inside the drum) along th...
35
Figure 3.3 Shows the change in mass fraction of the water (inside the cake) along the radius of
the drum. During the so...
36
6. REFERENCES
1 R.B.Bird , W.E.Stewart and E.N Lightfoot. Transport Phenomena. John Wiley & Sons,
New York. Second edit...
37
APPENDIX
MATLAB Solution
Solution for equation (1)
Chill roll call
function dT= chillroll(z,T)
qw=0.07;
clear all
clc
z...
38
Call Function
clear all
clc
zspan=[.01 2.5];
t0=4;
[z,T]=ode15s('chilltemp',zspan,t0);
plot(z,T);
xlabel('length (m)')
...
39
40
41
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Medha Singh Major Report (2)

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Medha Singh Major Report (2)

  1. 1. HEAT TRANSFER MAPPING OF DRUM FLAKER(CHILLROLLS) A MAJOR PROJECT REPORT Submitted in partial fulfillment of the requirements for the award of the degree of Master of Technology in CHEMICAL ENGINEERING (With Specialization in Process Design Engineering) By MEDHA SINGH DEPARTMENT OF CHEMICAL ENGINEERING UNIVERSITY OF PETROLEUM & ENERGY STUDIES, DEHRADUN UTTARAKHAND-248007, INDIA April -2016
  2. 2. 2 CANDIDATE’S DECLARATION I hereby declare that the work, which is being presented in this dissertation, Entitled “TOPIC’’, submitted in partial fulfillment of the requirement for award of the degree of Master of Technology in Chemical Engineering with the specialization in Process Design Engineering (PDE), is an authentic record of my own work carried out under the supervision of , Professor, Department of Chemical Engineering, University of Petroleum & Energy Studies, Dehradun and Guide 2 Name from industry. Date - 20th April, 2016 Place – Dehradun Medha Singh This is to certify that the above statement made by the candidate is correct to the best of my knowledge. Ms Ankitha R Kartha Professor, Dr Rajeshwar Mahajan NMB Plant Head Department of Chemical Engineering Hindustan Unilever Limited, University of Petroleum & Energy Studies Haridwar Uttarakhand Dehradun, Uttarakhand-248007 INDIA
  3. 3. 3 ACKNOWLEDGEMENT I am using this opportunity to express my gratitude to everyone who supported me throughout the course of this M.Tech project. I am thankful for their aspiring guidance, invaluably constructive criticism and friendly advice during the project work. I am sincerely grateful to them for sharing their truthful and illuminating views on a number of issues related to the project. I express my warm thanks to Ms Ankitha R Kartha for their support and guidance at Hindustan Unilever Ltd. I would also like to thank my project internal guide Dr .Rajeshwar Mahajan and all the people who provided me with the facilities being required and conductive conditions for my M.Tech project. Dated: April 20th 2016 Medha Singh Place:Dehradun
  4. 4. 4 ABSTRACT This report explores the parametric influence of spray cooling for thick walled metal drum with a thin layer of soap on it. Using the point-source depiction of a spray, an analytical model is derived to determine the heat and mass balance throughout the system. By setting boundary conditions for the sprayed portion a heat diffusion model is constructed. This mathematical model would help in heat transfer mapping and will provide a temperature curve for every point along the sprayed surface.
  5. 5. 5 Contents CANDIDATE’S DECLARATION.........................................................................2 ACKNOWLEDGEMENT.......................................................................................3 ABSTRACT..............................................................................................................3 LIST OF FIGURES .................................................................................................7 LIST OF TABLES ...................................................................................................8 NOMENCLATURE.................................................................................................9 1. INTRODUCTION ...........................................................................................11 2. LITERATURE REVIEW...............................................................................14 2.1 Working Principle and Parameters ............................................................15 2.2 Features Of The Drum Flaker(Chill rolls).................................................16 2.3 Specifications.................................................................................................16 2.3.1 The Drum ................................................................................................16 2.3.2 Coolant Circulation................................................................................17 2.3.3 Scrapper Assembly.................................................................................17 2.3.4 The Drum’s Material..............................................................................17 2.4 The Purpose Of Flaking ...............................................................................18 2.5 Parameter and the Tendency of the Product.............................................18 2.6 Types of nozzles.............................................................................................19 2.6.1 Hollow cone nozzles-Disc and core type...............................................20 2.6.2 Flat Fan Nozzles......................................................................................20 2.6.3 Floodjet nozzles.......................................................................................20 2.7 Cooling System In The Drum......................................................................20
  6. 6. 6 2.8 Applications...................................................................................................22 2.8.1 Food industry ..........................................................................................22 2.8.2 Fine Chemical..........................................................................................22 2.8.3 Healthcare ...............................................................................................22 3. MATHEMATICAL MODELLING.................................................................23 3.1 Methodology..................................................................................................23 3.2 Mathematical Modelling ..............................................................................23 3.3 Assumptions:.................................................................................................24 3.4 Mass balance for semi-differential control volume of water flowing in the z direction inside the pipe...................................................................................24 3.5 Heat Balance for the water flowing inside drum......................................25 3.6 Heat Transfer Through The Walls. ............................................................26 3.7 Mass Balance For The Cake .......................................................................27 3.8 Heat Transfer Balance (Cake).....................................................................28 4. SOLUTION........................................................................................................30 4.1 Final Equations To Solve .............................................................................30 5. RESULT AND DISCUSSION........................................................................33 6. REFERENCES ................................................................................................36 APPENDIX.............................................................................................................37
  7. 7. 7 LIST OF FIGURES No of Figure Name of Figure Page No 2.1 Drum flaker (Chill Roll) 15 2.2 Over of Drum flaker (Chill Roll) 16 2.3 Spray Nozzle Configuration 19 2.7.1 Full water type 21 2.7.2 Jacket type 21 2.7.3 Sprayer type 21 3.6.1 Assumed Flow Model (Heat and Fluid) Inside the Drum 27
  8. 8. 8 LIST OF TABLES No of Table Name of Table Page No 2.5.1 Table of Parameter and Tendency 18
  9. 9. 9 NOMENCLATURE Vz(z) mean z- velocity of liquid inside pipe Δ(z) thickness of the (“boundary” layer )of liquid in the drum qw flow rate of water being sprayed inside the drum (constant) 𝑚3/𝑠 𝑚2 𝑎𝑟𝑒𝑎 T(z) mean temperature of the water flowing inside the drum Twf temperature of the water being sprayed (constant) Rp inside radius of the pipe Q(z) heat flux (kj/m2 s) T0(z) temperature of the outer wall km thermal conductivity (kj/m-s-k) of the metal tm thickness of metal wall tcake thickness of cake ya*(r,z) mass fraction of water (a) in cake at (r,z) cpw specific heat of water Dab mass diffusivity of a through cake(constant) Ky mass transfer coefficient in the air film ya**(z) mass fraction at equilibrium with mass fraction at outer film yab mass fraction in atmosphere Tair temperature of air in the bulk Tcake temperature of cake kcake thermal conductivity of cake
  10. 10. 10 Greek symbols λw (Tcake) latent heat of vaporization of water at cake temperature r = (Rp +tm+tcake) ρ density of water
  11. 11. 11 1. INTRODUCTION Hindustan Unilever Limited (HUL) is India's largest Fast Moving Consumer Goods Company with a heritage of over 80 years in India and touches the lives of two out of three Indians. HUL works to create a better future every day and helps people feel good, look good and get more out of life with brands and services that are good for them and good for others. With over 35 brands spanning 20 distinct categories such as soaps, detergents, shampoos, skin care, toothpastes, deodorants, cosmetics, tea, coffee, packaged foods, ice cream, and water purifiers, the Company is a part of the everyday life of millions of consumers across India. Its portfolio includes leading household brands such as Lux, Lifebuoy, Surf Excel, Rin, Wheel, Fair & Lovely, Pond’s, Vaseline, Lakmé, Dove, Clinic Plus, Sunsilk, Pepsodent, Closeup, Axe, Brooke Bond, Bru, Knorr, Kissan, Kwality Wall’s and Pureit. The Company has over 16,000 employees and has an annual turnover of INR 30,170 crores (financial year 2014 – 15). HUL is a subsidiary of Unilever, one of the world’s leading suppliers of fast moving consumer goods with strong local roots in more than 100 countries across the globe with annual sales of €48.4 billion in 2014. Unilever has 67.25% shareholding in HUL. In 1931, Unilever set up its first Indian subsidiary, Hindustan Vanaspati Manufacturing Company, followed by Lever Brothers India Limited (1933) and United Traders Limited (1935). These three companies merged to form HUL in November 1956; HUL offered 10% of its equity to the Indian public, being the first among the foreign subsidiaries to do so. Unilever now holds 67.25% equity in the company. The rest of the shareholding is distributed among about three lakh individual shareholders and financial institutions. The erstwhile Brooke Bond's presence in India dates back to 1900. By 1903, the company had launched Red Label tea in the country. In 1912, Brooke Bond & Co. India Limited was formed. Brooke Bond joined the Unilever fold in 1984 through an international acquisition. The erstwhile
  12. 12. 12 Lipton's links with India were forged in 1898. Unilever acquired Lipton in 1972, and in 1977 Lipton Tea (India) Limited was incorporated. Pond's (India) Limited had been present in India since 1947. It joined the Unilever fold through an international acquisition of Chesebrough Pond's USA in 1986. Since the very early years, HUL has vigorously responded to the stimulus of economic growth. The growth process has been accompanied by judicious diversification, always in line with Indian opinions and aspirations. The liberalisation of the Indian economy, started in 1991, clearly marked an inflexion in HUL's and the Group's growth curve. Removal of the regulatory framework allowed the company to explore every single product and opportunity segment, without any constraints on production capacity. Simultaneously, deregulation permitted alliances, acquisitions and mergers. In one of the most visible and talked about events of India's corporate history, the erstwhile Tata Oil Mills Company (TOMCO) merged with HUL, effective from April 1, 1993. In 1996, HUL and yet another Tata company, Lakme Limited, formed a 50:50 joint venture, Lakme Unilever Limited, to market Lakme's market-leading cosmetics and other appropriate products of both the companies. Subsequently in 1998, Lakme Limited sold its brands to HUL and divested its 50% stake in the joint venture to the company. HUL formed a 50:50 joint venture with the US-based Kimberly Clark Corporation in 1994, Kimberly-Clark Lever Ltd, which markets Huggies Diapers and Kotex Sanitary Pads. HUL has also set up a subsidiary in Nepal, Unilever Nepal Limited (UNL), and its factory represents the largest manufacturing investment in the Himalayan kingdom. The UNL factory manufactures HUL's products like Soaps, Detergents and Personal Products both for the domestic market and exports to India.
  13. 13. 13 1.1 Motivation for the work In Hindustan unilever Limited, in NMB plant (soap plant). The drum flaker is used primarily to process chemical and pharmaceutical products. The drum flaker transforms a molten product into a solid. The process that takes place is a solidification and/or crystallisation process. The product begins to solidify the moment it comes into contact with the cold, rotating drum. After one revolution the completely solidified layer is removed from the drum by a knife and typically breaks into easy to handle flakes. It is suitable to produce soap flakes at a temperature of 38- 40°C approx. from liquid soap at 80-90°C. But the problem arises when the drum’s temperature increases upto 35°C, which has to be maintained at 15-17°C. I took this project as this is a real life industrial problem, where I can apply my basics and logic of transport phenomena to build a mathematical model of this whole process. The objectives of this study are as follows To develop a mathematical model. To examine the effects of operating parameters. Development of computer programme to solve the mathematical model in MATLAB .
  14. 14. 14 2. LITERATURE REVIEW The drum flaker is used primarily to process chemical and pharmaceutical products. However, more and more applications for these machines are also being found in the food industry. The closed design is ideally suited for processing of toxic or offensively smelling products. With the drum flaker, a molten product is converted into a solid form. A thin layer of the liquid product adheres to the outside of the rotating, internally cooled drum in a continuous process. Heat is extracted from the product by contact with the cooled drum surface, and the product solidifies and cools to the required final temperature. A stationary knife removes and breaks up the solidified layer. The required flake size is achieved by controlling circumferential speed, layer thickness, and knife angle. Careful design ensures optimum use of the drum surface area to maximize capacity at the chosen operating conditions. The drum flaker is primarily used to produce flakes, but there are also ways of converting your product into easily manageable pastilles or prills. It has a two, counter-rotating stainless steel rolls equipped with a sprayer with nozzles from which cold water is sprayed around the inner surface of the drum, to give a uniform heat transmission. The drums are equipped with rotary joints for proper water distribution and consistent water recycling through siphon. Separate motor-reducers drive each drum and the product discharge conveyor These independent drives allow easy maintenance and flexibility of operation. Liquid soap slurry falls on the rolls that spread it evenly among them, in a 0,3 — 0,5 mm thick layer that is quickly cooled down and solidified.
  15. 15. 15 Product is then scraped off the rolls, by the doctor blades, in form of flakes and fall onto a discharge conveyor belt. It is suitable to produce soap flakes at a temperature of 38-40°C approx. from liquid soap at 80-90°C. To maintain this surface temperature a detailed study has been done on the structure and the whole process. 2.1 Working Principle and Parameters A detailed literature review has been done to understand the process of chill roll (drum flaker). Liquid soaps slurry falls on the rolls that spread it evenly among them, in a 0.3 — 0.5 mm thick layer that is quickly cooled down and solidified. Product is then scraped off the rolls, by the doctor blades, in form of flakes and fall onto a discharge conveyor belt. It is suitable to produce soap flakes at a temperature of 38-40°C approx. from liquid soap at 80-90°C. Figure 2.1 Drum flaker (chill rolls)
  16. 16. 16 Figure 2.2 Overview of Drum Flaker (Chill rolls) 2.2 Features of the Drum Flaker (Chill rolls) The Drum Flaker enables processing of molten products into excellent quality flakes. The well- considered concept of the drum flaker has led to a number of features: • Compact unit, little floor space required; • Completely closed cooling system, absolutely no cross-contamination between cooling medium and product; • Low operational and maintenance costs • Gastight enclosures • Easy inertisation of the process; • Unit designed with good access for maintenance and cleaning • Construction material ranges from carbon steel to various grades of stainless steel, Haste alloy, etc. 2.3 Specifications 2.3.1 The Drum Cooling drums are available in a choice of materials selected in accordance with the chemical properties and adhesion ability of the product to be processed. The materials range from fine grain cast iron and carbon steel to Haste alloy, etc. The drums are specially designed for maximum geometric stability. Distortion due to differential temperature gradient or mechanical forces is
  17. 17. 17 impossible in normal operation. The drum design ensures equal heat transfer over the entire drum surface, ensuring uniform flake size distribution. For products with poor adhesion to metal surfaces, drums with special grooved surfaces are available. 2.3.2 Coolant Circulation When liquid refrigerants are used, these are sprayed by a central spray tube over the internal drum surface. The turbulent flow ensures maximum heat transfer over the full drum surface, including over shell ends and heads. Consequently, an equal temperature is guaranteed over the total length of the drum, resulting in uniform flakes with low fines content. The liquid is siphoned off from the lowest part in the drum to avoid an accumulation of refrigerant there. The siphoning action is assisted by some slight overpressure inside the drum. The entire system, made of stainless steel, is easily detachable without disassembly of the drum. This design enables feed and discharge of the refrigerant through one and the same shaft, while the other shaft is used for the drive. Moreover it offers excellent accessibility to the inside of the drum for cleaning and inspection. Some options are available, such as refrigeration through direct evaporation of Freon or ammonia. 2.3.3 Scrapper Assembly Rigid construction designed to ensure a uniform pressure against the drum over the full length and to eliminate vibrations. The knife pressure is effected and controlled by means of a pneumatic pressing system. For entirely enclosed machinery, the knife’s pressing system is located outside the process environment. Scraper knives are available in a range of materials, from steel to technical plastics. The flake size can be determined in advance by the choice of scraper system. 2.3.4 The Drum’s Material The drums are mostly made of stainless steel. Besides the choice of many types of stainless steel, chromium-plated, Haste alloy, or cast iron drums are also possible. The exact choice will depend on your product, the work site environment, available space, and the process to be performed. The result is a durable drum with high dimensional stability and uniform heat distribution over its entire surface.
  18. 18. 18 2.4 The Purpose of Flaking The purpose of flaking is to cool and solidify fusion liquid. The solidified product is shaped thin flake. The flake is capable of conveyance, safekeeping, transportation, and discharging. 2.5 Parameter and the Tendency of the Product When the following parameter is changed, it’s possible to change the several conditions of the flake. ・Rotation speed of the drum ・Temperature of the cooling water ・Clearance of the drum and the levelling roll Table 2.5.1 Table of Parameter and Tendency
  19. 19. 19 Fig. 2.3 Spray nozzle configuration This is a spray nozzle arrangement along the axis of a cylindrical thick-walled tube. Water is fed through an axial supply channel that is fitted with radial oriented spray nozzles. The supply channel and nozzles are fully retractable. They are inserted initially along the axis after the heated part is removed from the furnace to initiate the quench, and retracted upon completion of the quench. To maximize spray impact area and ensure both effective and predictable spray distribution, the nozzles are positioned such that the spray impact zones along the inner curved surface of the tube are tangent to one another with no spacing or overlap. This arrangement limits the nozzle-to-surface distance to less than the radius of the tube. 2.6 Types of nozzles A spray nozzle is a precision device that facilitates dispersion of liquid into a spray. Nozzles are used for three purposes  To distribute a liquid over an area.  To increase liquid surface area.  To create impact force on a solid surface.
  20. 20. 20 A wide variety of spray nozzle applications use a number of spray characteristics to describe the spray. 2.6.1 Hollow cone nozzles-Disc and core type These are used primarily where plant foliage penetration is essential for effective insect and disease control, and where drift is not a major consideration. At pressures of 40 – 8- psi hollow cone nozzles give excellent spray coverage to the undersides of reduces penetration correspondingly 2.6.2 Flat Fan Nozzles Flat Fans are most commonly used spray nozzles. They produce a range of droplet sizes from coarser to finer depending on pressure which makes them suitable to use. Flat Fan provides a very good spray distribution over a wide range of pressures. 2.6.3 Flood jet nozzles These are ideal for high application rates and speeds, because they produce a wide-angle, flat fan pattern. Operating flood-jet nozzles at 5-25 psi minimizes drift, but pressure changes critically affect the width of the spray pattern. Generally, the spray generated by the flood jet is not as uniform as the flat-fan type. 2.7 Cooling System in the Drum There are 3 ways of cooling system of drum flake 1- Cooling water is filled in the lower half inside the drum. The structure is easiness, but because the water which became lukewarm is left, the cooling ability falls.
  21. 21. 21 . Figure 2.7.1: Full water type 2- Cooling water flows in the jacket. Without being left, lukewarm water will be discharged immediately Figure 2.7.2: Jacket type 3- Cooling water is sprayed in the drum. The ability is high, but refuse is sometimes jammed into a sprayer nozzle. Compressed air is needed. Figure 2.7.3: Sprayer type
  22. 22. 22 2.8 Applications Chilled roll flaker is a multi-purpose drying device. It is generally used in industries where the product is temperature sensitive. Some examples are stated below 2.8.1 Food industry 1. Cheese 2. Chocolate 3. Dough 4. Vegetables 2.8.2 Fine Chemical 1. Fatty acids 2. Oleo chemicals 3. Phtalic Anhydride 4. Maleic Anhydride 5. Calcium Chloride 6. Caprolactam 7. Resins 8. Bisphenol A 9. Sulphur 2.8.3 Healthcare 1. Stearate 2. Soaps
  23. 23. 23 3. MATHEMATICAL MODELLING 3.1 Methodology The objective of my project is to study and Construct a heat transfer model of the system by heat and mass balance of the overall system to trouble shoot the problem of overheating the surface .Modelling requires understanding of engineering systems. – By observation and experiment. – Theoretical analysis and generalization. – Assumptions. Modelling of this system includes taking a small section and forming differential equation for it. Then solving it with some boundary conditions, to get heat and mass balance equations which can be solved using MATLAB to get temperature profile, velocity profile and of the system. 3.2 Mathematical Modelling Mathematical modelling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application. Mathematical modelling  is indispensable in many applications  is successful in many further applications  gives precision and direction for problem solution  enables a thorough understanding of the system modelled  prepares the way for better design or control of a system  allows the efficient use of modern computing capabilities
  24. 24. 24 Learning about mathematical modelling is an important step from a theoretical mathematical training to an application-oriented mathematical expertise, and makes the student fit for mastering the challenges of our modern technological culture. 3.3 Assumptions: • Constant flow of fluid • Stationary system. • constant density and viscosity • Heat transfer only through conduction and convection • flow in only z direction 3.4 Mass balance for semi-differential control volume of water flowing in the z direction inside the pipe. Steady state Vz2πRp. δ(z) + qw. dz2πRp = Vz2πRp. δ(z) + qw. dz + 2πRp(Vzδ)dz Or 2πRp. d(Vz.δ) dz . dz = qw. dz2πRp Or 2πRp. d(Vz. δ) dz = qw2πRp After cancelling all the common terms we get d(Vz.δ) dz = qw (3.1)
  25. 25. 25 Integrate above equation over z Vzδ = qwz + C1 Boundary conditions At z = 0, Vz = δ = 0 therefore C1 = 0 We get Vzδ = qwz (3.2) 3.5 Heat Balance for the water flowing inside drum. Cold water is being sprayed inside the drum, spreads evenly inside at the wall of the drum. This flowing water takes heat from the cake (falling on the outer surface of the drum) through the metal wall. Water which are spraying inside the drum gets collected at the bottom forming a thick layer which is increasing due to syphoning process. As this whole process is a continuous process i.e at the same time the cold water is coming in and the hot water is going out. Considering Heat is transferring from outside to inside through the walls of the drum (metal wall) to water. ρw[Vz. 2πRp. δ]Cpw + ρw[2πRp. dz]qw. Cpw. Twf = ρw 2πRpCpwVz. Δ.T + ρw 2πRpCpw d dz (Vz δT)dz -Q 2πRpdz After cancelling all the common terms we get, qwTwf = d dz (VzδT) + Q Cpwρw (3.3) From equation (2) substitute the value of Vzδ
  26. 26. 26 qw. Twf = d dz (qwz. T) + Q(z) Cpwρw Or qw. Twf = qw d dz (z. T) + Q(z) Cpwρw Or qw. Twf = qw [z dT dz + T. 1] + Q(z) Cpwρw Or qw (Twf − T) = qw z dT dZ + Q(z) Cpwρw (3.4) 3.6 Heat Transfer through the Walls. Heat is transferred from cake to the metal wall followed by conduction process. Fourier’s law of conduction Q(z) = Km (T(z) − T0) tm (3.5)
  27. 27. 27 Fig 3.6.1 Assumed Flow Model Inside The Drum (Heat and Fluid) 3.7 Mass Balance For The Cake Assuming only radial direction a = water b = solid ya*= fraction of a at (r,z) In this process when hot soap slurry falls on a drum , it contains some moisture. After several rotations this moisture diffuses into the atmosphere and then the mixture becomes solid. A small portion from cake is considered let’s say r , r+dr in radial direction through which liquid is diffusing into the atmosphere.
  28. 28. 28 Mass Balance (steady state) In-Out=0 - -Dab.2πrdz. dya ∗ dr -[- Dab.2πrdz. dya ∗ dr - Dab.2πdz. d dr (r . dya ∗ dr )dr ]=0 or d dr [r dya ∗ dr ] = 0 (3.6) Boundary Conditions At r = Rp + tm dya ∗ dr = 0 At r = Rp + tm + tcake −Dab. dya ∗ dr = −ky(ya ∗∗ −yab) 3.8 Heat Transfer Balance (Cake) Heat transfer balance includes conduction and convection through the surface −kcake. 2πrdz. Kcake. 2πdz (r dTcake dr ) − Kcake. 2πdz d dr (r dTcake dr ) dr = 0
  29. 29. 29 Or d dr [r dTcake dr ] = 0 (3.7) Boundary conditions At r = Rp + tm Tcake = T0 At r = Rp + tm + tcake −Kcake. 2π(Rp + tm + tcake)dz dTcake dr hair. 2π(Rp + tm + tcake )dz(Tair − Tcake) = λw(Tcake). Ky(Ya ∗∗ − Yab). 2π(Rp + tm + tcake) or hair(Tair − Tcake) − kcake dTcake dr = λw(Tcake). Ky(Ya ∗∗ − Yab) (3.8)
  30. 30. 30 4. SOLUTION 4.1 Final Equations To Solve Vzδ = qwz (3.2) qw (Twf − T) = qw z dT dZ + Q(z) Cpwρw (3.4) Q(z) = Km (T(z) − T0) tm (3.5) d dr [r dya ∗ dr ] = 0 (3.6) Boundary Conditions r = Rp + tm dya ∗ dr = 0 r = Rp + tm + tcake −Dab. dya ∗ dr = −ky(ya − yi) d dr [r dTcake dr ] = 0 (3.7)
  31. 31. 31 Boundary Conditions At r = Rp + tm Tcake = T0 At r = Rp + tm + tcake hair(Tair − Tcake) − Kcake dTcake dr = λw(Tcake). Ky(ya ∗∗ − yab) After rearranging above equation Vzδ = qwz (4.1) dT dz = − ( km ztmCpwρwqw + 1 z ) T + kmT0 ztmCpwρw + Twf z (4.2) dya ∗ dr = ky Dab (ya ∗∗ − yab) (4.3) ya ∗ = B − B−B∗ ln( r1 r2 ) . ln(r/r1 ) dTcake dr = C1 r (4.4)
  32. 32. 32 At r = r1 Tcake = T0 At r = r2 −k dTcake dr = λw. Tcake. ky(ya ∗∗ − yab) − hair(Tair − Tcake) C1 = − r2 k [λw. Tcake. ky(ya ∗∗ − yab) − hair(Tair − Tcake)] = (Tcake−T0) lnr1
  33. 33. 33 5. RESULT AND DISCUSSION This study examined the problem of spray cooling thick walled metal. An analytical model was constructed to determine the temperature variation throughout the curved inner wall of the drum, the computational model provides curves for every point within the sprayed surface of the wall. This study examined the problem or increasing the surface temperature of drum flaker(chill roll).Which could be improved by adjusting the flow rates, siphoning process, by adjusting blades, by adjusting nozzle to inside surface length, by adjusting size of nozzle, spray angle. Figure3.1 Shows straight line which means the velocity and the thickness of the water flowing inside the drum increasing along the length of the drum. Figure 3.1 velocity profile of water 0 0.5 1 1.5 2 2.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Length (m) (Vz*del)(m2/s)
  34. 34. 34 Figure 3.2 Shows the change in temperature (T) profile (mean temperature of the water flowing inside the drum) along the length (L) of the drum. It first gradually increases and then becomes constant. This is the temperature which has to be maintained below 6 ͦCelsius throughout the process. Because this temperature causes the problem of overheating of the drum. If this temperature remains constant below 6 ͦ Celsius automatically the temperature of the outer surface of the drum will gradually decrease and could be maintained at the desired temperature. Figure 3.2 temperature (T) profile of water 0 0.5 1 1.5 2 2.5 4 4.005 4.01 4.015 4.02 4.025 4.03 4.035 4.04 4.045 length (m) temperature(T)C
  35. 35. 35 Figure 3.3 Shows the change in mass fraction of the water (inside the cake) along the radius of the drum. During the solidification process the amount of water in the cake gets evaporate into the atmosphere. Figure 3.3 Concentration profile of water in the cake 1.5 1.505 1.51 1.515 1.52 1.525 1.53 1.535 0.175 0.18 0.185 0.19 0.195 0.2 radius (m) massfractionofwater(ya)
  36. 36. 36 6. REFERENCES 1 R.B.Bird , W.E.Stewart and E.N Lightfoot. Transport Phenomena. John Wiley & Sons, New York. Second edition , 2002 2 Robert H Perry , Don W Green. Perry’s Chemical Engineers Handbook. Mc Graw Hill , New York. Seventh edition , 1999 3 N. Mascarenhas, I. Mudawar / International Journal of Heat and Mass Transfer 55 (2012) 2953–2964
  37. 37. 37 APPENDIX MATLAB Solution Solution for equation (1) Chill roll call function dT= chillroll(z,T) qw=0.07; clear all clc zspan=[0 2.5]; a0=0; [z,a]=ode45('chillroll',zspan,a0) plot(z,a) xlabel('Length (m)'); ylabel('(Vz*del) (m^2/s) '); Solution for equation (2) function dT= chilltemp(z,T) km=16.2;%W/mK tm=.03;%m cpw=4120;%kJ/kgK rhow=1000;%kgm^-3 qw=0.07;%m^3/h T0=15;%K Twf=4;%K % dT=-(km/(z*tm*cpw*rhow*qw)+1/z)*T+(km*T0)/(z*tm*cpw*rhow*qw)+Twf/z; dT=((km/(tm*cpw*rhow*qw))-1)*(T/z)+(((km*T0)/(z*tm*cpw*rhow*qw))+Twf/z);
  38. 38. 38 Call Function clear all clc zspan=[.01 2.5]; t0=4; [z,T]=ode15s('chilltemp',zspan,t0); plot(z,T); xlabel('length (m)') ylabel('temperature(T)C') Solution for equation (3) Mass balance function dya=massbalance(r,ya) ky=8*10^-6; Dab=.3*10^-6; yaeq=.04; yab=.07; dya=((ky/Dab)*(yaeq-yab)); clear all; clc; zspan=[1.503 1.533]; [r,ya]=ode45('massbalance',zspan,.2); plot(r,ya) xlabel('radius (m)'); ylabel('mass fraction of water(ya)');
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