##
Soğuk Hava Jetinin Sayısal Olarak Modellenmesi

Soğuk Hava Jetinin Sayısal Olarak Modellenmesi

##### Dosyalar

##### Tarih

1998

##### Yazarlar

Sertan, Uğur

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Bu çalışmada, soğuk hava jetinin hacim içerisindeki yayılımı ve jetin ayrılma mesafesinin sebepleri incelenmiştir. Çalışılan oda geometrisi, yatay duvar üzerinde yukarıdan zorlanmış hava girişi ve bu giriş karşısında aşağıdan doğal hava çıkışı olacak şekilde dizayn edilmiştir. Böyle bir durum pratik haricinde, kapalı bir ortamdaki zorlanmış taşınımı ifade etmekte olduğu izlenimi verse de, incelenen akışın tek başına zorlanmış ya da doğal taşınım olduğu söylenemez. Bu nedenle türbülansa geçiş parametreleri incelenmelidir. Bölüm l'de soğuk hava jeti tanımı yapılırken, çeşitleri de belirtilmiştir. Aynı zamanda literatür taraması da yapılıp, konuyla ilgisi olan kısımlar teze dahil edilmiştir. Bölüm 2'de ise, soğuk hava jetinin ayrılması, modellemesi ve çeşitli parametreler gözönüne alınarak, jetin ayrılma mesafesi ampirik ifadelerle verilmiştir. Bölüm 3 'de, zorlanmış taşınım ve türbülansla ilgili yönetici denklemler ortaya çıkartılmıştır. Bu denklemler kullanılarak bilinmeyen bağımlı değişkenlerin çözülmesi denklem sayısını eşitleyerek sağlanmıştır. Bölüm 4, tamamen sayısal çözümü anlatan bir kısımdan ibarettir. Burada, ayrıklaştırılmış denklemlerle, diferansiyel denklemlerin çözümüne gidilmiştir. Bölüm 5, hacmin geometrisini, sınır şartlarını ve sonuçlarını veren son bölümdür. Bu bölümde sayısal olarak bilgisayarda çözülen değerler, farklı boyutsuz sayılarla karşılaştırılarak, hangi parametrenin ne yönde sonuçlara etki yaptığı tartışılmıştır. Ayrıca incelenen oda içerisindeki hız, sıcaklık dağılımları ile akış çizgileri elde edilmiş, soğuk hava jetinin oda boyunca nasıl bir karakter izlediği söylenebilmektedir.

Cold air distribution systems can offer greater energy, load, and space savings relative to competing systems due to smaller duct and fan size requirements. With a cold air distribution system, both the supply air temperature and flow rate are lower than in conventional systems for the same cooling load. In this study a simple room was examined for analysing the cold air jet, and at the end resulting values were compared with some characteristic parameters, which are called as nondimensionel numbers. During the year, the jet characteristics examined were the momentum, tempreture and velocity profiles of attached ceiling jets, and the governing parameters and location of jet seperation. In the first Chapter of this thesis, we defined firstly the cold air jet which is very simple explained. Cold air jets are colder than the room temperature. Therefore, when the jet thrown from the diffuser, it will drop some distance later. This is going to be discuss in the following parts. The jets thrown from the diffuser, have different characteristics. This means that, cold air jets have different characteristics when they are supplied from different types of diffuser, or under different conditions (initial air temperature, room geometry and size, supply direction, etc.). When the temperature of supplying air jet is equal to the temperature of the air in the room, the jet is called an "isothermal jet". When there is temperature difference between the incoming air jet and the room air, the jet is called a "nonisothermal jet". When the jet discharging into a large open space (not influenced by walls and ceiling), the jet is called a "free jet", whereas when the incoming air jet is attached to a ceiling or wall, it is called an "attached jet". It has been shown that by experimental and computational results, existing cold air jets has two-dimensional characteristics. Because of this reason, we can analyze the cold air jets on Cartesian coordinate both in numeric and analytic methods. In the second Chapter, the behavior of cold air jets, specifically jet throw and separation, for application to cold air distribution system were examined. It is presented a simple model of jet separation, develop throw(velocity - distance) equations for cold air jets. The results show how the jet mixing and separation are related to the momentum and the cooling capacity of the jet. Also, there is a nondimensionel parameter which is called Archimed number. Definition of Archimed number is the ratio of the buoyancy to inertia forces of the supply jet. A direct relationship between the separation point and the Archimed number is developed. However, there is a need to know more about how conventional diffüsers perform in a cold air application as the room- jet temperature difference is increased, and mass flow rate is decreased. For thermal comfort, there is a need to have more xu information about the relationship between the throw (distance to a specified) and the separation of the jet. The jet throw is usually defined as the distance from the diffuser outlet to the location where centerline velocity has decreased to a specified terminal velocities. For a cold air jet, through the flow, four main region can be recognized. Zone 1 : A short zone near the orifice where the maximum velocity is very closed to the discharged velocity. Zone 2 : A transition zone where the maximum velocity varies inversely as the square root of the distance from the diffuser. Zone 3 : An extensive zone (called fully established turbulent zone) where the maximum velocity varies inversely as the distance from the diffuser. Zone 4: A terminal zone where the jet decays rapidly into large scale turbulence or eddies. In application of air supply jets to ventilation and air conditioning the important parts of the jet are zones 2 and 3. Zone three of the jet expansion is the characteristic zone for most diffusers. In this zone the flow behaves as if it was generated by a point source. Centerline velocities have always been considered the most important jet characteristic for ventilation design, as they are the basis for the throw evaluation. For isothermal jets in the centerline velocities definition, the density ratio is negligible. The centerline velocity in the jet at a certain distance x from the outlet depends on the jet momentum.lt is seen that, jets from different sizes of the same type of diffuser will produce the same velocities at a given distance from the diffuser, if the jets have the same momentum. For the supply of cold air, temperature at the jet centerline must also be considered. The jet outlet velocity will need to be increased by the same proportion that the temperature difference is increased. The velocity increase can be accomplished by a decrease in the diffuser area. Some cases of the increased outlet velocity are increased energy demand for the jet, in proportion to the increased velocity. On the other hand, using a force balance applied to a simple model, the separation distance can be related to the jet momentum, density, and room cooling load. Separation occurs when the negative buoyancy force, due to the temperature difference in the jet is larger than the attachment force The upward attachment force, F^^ on a segment of the jet is proportional to M/x, so Coanda force decreases with increasing x. From Archimeds principle, the downward gravitational force on a segment of the jet is proportional to the volume and temperature difference of the jet. This shows that the gravitational force goes as Qx/M!/2, increasing linearly with distance. At any distance x, the ratio between the Coanda force and the gravitational force is proportional to momentum flow, and inversely proportional to cooling capacity. xni The jet separates from the ceiling at some critical ratio of Coanda and gravitational forces. Finally, Archimed number which is the ratio of buoyancy to inertia forces can be expressed below. Ar = g3A60 (1) This nondimensional number will be discussed and compare with the other results later on. In Chapter 3, laminar and turbulence flow are explaning for incompressible fluids. First of all, it was looked the governing equations both for laminar and turbulence cases. If we quik look the equations below, we can understand the other part of study. Laminar Flow Case (governing equations-cartesian coordinates) Mass Conservation 1- - = 0 dx dy (2) Momentum Equations: du du dxx ÖP \dt ox dy) dx rd2xx ] d2^.dx2+dy2) (3) (8w 8v d\] ÖP (d2v dV - + u - + v - = + u - r + - r Ut dx dy) dx \dx2 dy2) (4) Energy Equation dl dl 8T + u - + v- ^2 a2T azT 2 tA +. dt dx dyj p cp v d x dy J (5) Turbulence Flow Case (governing equations-cartesian coordinates) Mass Conservation : p - * - - = 0 dx: (6) Momentum Equation : p au± cm at Jax; ?iJ dx.. au dx. d? 'dx, +P& (7) Energy Equation : P rdi TT di) Kdt 'dx.j (\i dT dX: iV Prax.- -pUiT (8) These equations help us to predict the unknown variable dependent u, v, T, k, e values.For turbulent case there are more equations we need.If we briefly explain these equations we must write them. xiv In order to solve enclosed problem for turbulance the common method from Boussinessq is called eddy viscosity aproximation. According to this method Reynold stress are written below: -UjUj = v, aiL dV \ J-+ dx; dXi 3,J (9) In this equation turbulent viscosity, v, and turbulent Prandtl number at are properties of flow, not fluid. Modelling turbulence is widely investigated subject especially in the last two decades, as a result of rapid developmants in computers and numerical methods. One of these models, called "k-e" model, is very comman, and also forms the fundemantals of the model developed in this thesis. In Chapter 3, fundemantals of k-e model are explained. In this method, the following governing equations are used : dk TT Ök Ö fv, dk] - + Uj = - dt dx; oXj ^ak d x.J + vt fau, sUjI dxs ?+? OX, au; 6X; -8 (10) de ds ~dt+ 'axT vt 5 s dxAGa dx,. + v, 1 + ? ox, dxj ?^-^T-^ıT Ol) ox, k k and solved. The reason for accepting turbulent kinetic energy as the velocity scale and the definition of turbulent kinetic energy is given. Two additional transport equations for turbulent kinetic energy k, and the dissipation of turbulent kinetic energy are solved together with the governing equations. These equations and emprical constants and coefficients are presented in the thesis. Then the insufficiency of this model in flows with low Reynolds numbers are described. Since the analytical solution is imposible, the difficulties for solving k-e turbulent model should be overcome. For that reason modified models developed by various researchers. These developed models are called 'low Reynolds number models" For this model, the differantial equations for turbulent kinetic energy and the dissipation of turbulant kinetic energy are shown below. Dk Dt i a PaXj + "? ak OX: + u.au. P 5XJ f au; au^ L + i ax= ax; -e + D (12) Ds 1 a Dt paxj + fiCel e u,, aUj k p ax= au} au^ -+ IdXj dxj k (13) v, = f"c" - (14) xv Numerical solution consists of discretization by finite difference method and an iterative solution by the means of the code developed in this study. The general form of the dicretized differential equations, variable dependent and independent sources, the flow diagram of the code and the grid distribution are present in the Chapter 4. Grids are generated in such a way that dimensions of the control volumes are very tiny near the walls and they become coarser towords the core region. The grid field is a staggered one that means vectorial dependent variables have been defined at the grid points. In the code, the Power Law Differencing Scheme Method, PLDS, for the convection terms and Semi - Implicit Pressure Linkage Method, SIMPLE, for the pressure correction have been used. Both of these methods are discribed briefly. Also, underrelaxation equations and factors for every variable, and the convergence criteria are the subjects discussed. General boundry conditions are given in Chapter 5. At the walls, velocity, pressure, and turbulent kinetic energy equal to zero; dissipation of kinetic energy, temperature at the horizontol walls are all zero gradiends, while the temperature values at the vertical walls are taken as constant. At inlet, velocity, temperature variables are constant, turbulent kinetic energy is a function of inlet velocity and dissipation of turbulant kinetic energy is a function of the turbulant kinetic at the inlet, while the pressure gradient equals to zero. At outlet section all the dependent variables, except pressure, equal to constant fluxes while the pressure has zero gradient. The results we get, compared between the nondimensional numbers Ar, Re ve Fr. Fr = - 7 r- (Froud number) (15) gs T. V T. oda- US Re = - (Reynolds number) (16) Firstly, when increasing Re, the seperation distance reducing. This means that, since Re number is a charecteristics of turbulence, that result is obvious. Similarly, as the Ar number is increasing, the seperation distance is reducing. It means, the effect of buoyancy into turbulence is very strong. In addition, as the Fr decreasing, seperation distance decreasing as well. This result, completely opposite with the second result.

Cold air distribution systems can offer greater energy, load, and space savings relative to competing systems due to smaller duct and fan size requirements. With a cold air distribution system, both the supply air temperature and flow rate are lower than in conventional systems for the same cooling load. In this study a simple room was examined for analysing the cold air jet, and at the end resulting values were compared with some characteristic parameters, which are called as nondimensionel numbers. During the year, the jet characteristics examined were the momentum, tempreture and velocity profiles of attached ceiling jets, and the governing parameters and location of jet seperation. In the first Chapter of this thesis, we defined firstly the cold air jet which is very simple explained. Cold air jets are colder than the room temperature. Therefore, when the jet thrown from the diffuser, it will drop some distance later. This is going to be discuss in the following parts. The jets thrown from the diffuser, have different characteristics. This means that, cold air jets have different characteristics when they are supplied from different types of diffuser, or under different conditions (initial air temperature, room geometry and size, supply direction, etc.). When the temperature of supplying air jet is equal to the temperature of the air in the room, the jet is called an "isothermal jet". When there is temperature difference between the incoming air jet and the room air, the jet is called a "nonisothermal jet". When the jet discharging into a large open space (not influenced by walls and ceiling), the jet is called a "free jet", whereas when the incoming air jet is attached to a ceiling or wall, it is called an "attached jet". It has been shown that by experimental and computational results, existing cold air jets has two-dimensional characteristics. Because of this reason, we can analyze the cold air jets on Cartesian coordinate both in numeric and analytic methods. In the second Chapter, the behavior of cold air jets, specifically jet throw and separation, for application to cold air distribution system were examined. It is presented a simple model of jet separation, develop throw(velocity - distance) equations for cold air jets. The results show how the jet mixing and separation are related to the momentum and the cooling capacity of the jet. Also, there is a nondimensionel parameter which is called Archimed number. Definition of Archimed number is the ratio of the buoyancy to inertia forces of the supply jet. A direct relationship between the separation point and the Archimed number is developed. However, there is a need to know more about how conventional diffüsers perform in a cold air application as the room- jet temperature difference is increased, and mass flow rate is decreased. For thermal comfort, there is a need to have more xu information about the relationship between the throw (distance to a specified) and the separation of the jet. The jet throw is usually defined as the distance from the diffuser outlet to the location where centerline velocity has decreased to a specified terminal velocities. For a cold air jet, through the flow, four main region can be recognized. Zone 1 : A short zone near the orifice where the maximum velocity is very closed to the discharged velocity. Zone 2 : A transition zone where the maximum velocity varies inversely as the square root of the distance from the diffuser. Zone 3 : An extensive zone (called fully established turbulent zone) where the maximum velocity varies inversely as the distance from the diffuser. Zone 4: A terminal zone where the jet decays rapidly into large scale turbulence or eddies. In application of air supply jets to ventilation and air conditioning the important parts of the jet are zones 2 and 3. Zone three of the jet expansion is the characteristic zone for most diffusers. In this zone the flow behaves as if it was generated by a point source. Centerline velocities have always been considered the most important jet characteristic for ventilation design, as they are the basis for the throw evaluation. For isothermal jets in the centerline velocities definition, the density ratio is negligible. The centerline velocity in the jet at a certain distance x from the outlet depends on the jet momentum.lt is seen that, jets from different sizes of the same type of diffuser will produce the same velocities at a given distance from the diffuser, if the jets have the same momentum. For the supply of cold air, temperature at the jet centerline must also be considered. The jet outlet velocity will need to be increased by the same proportion that the temperature difference is increased. The velocity increase can be accomplished by a decrease in the diffuser area. Some cases of the increased outlet velocity are increased energy demand for the jet, in proportion to the increased velocity. On the other hand, using a force balance applied to a simple model, the separation distance can be related to the jet momentum, density, and room cooling load. Separation occurs when the negative buoyancy force, due to the temperature difference in the jet is larger than the attachment force The upward attachment force, F^^ on a segment of the jet is proportional to M/x, so Coanda force decreases with increasing x. From Archimeds principle, the downward gravitational force on a segment of the jet is proportional to the volume and temperature difference of the jet. This shows that the gravitational force goes as Qx/M!/2, increasing linearly with distance. At any distance x, the ratio between the Coanda force and the gravitational force is proportional to momentum flow, and inversely proportional to cooling capacity. xni The jet separates from the ceiling at some critical ratio of Coanda and gravitational forces. Finally, Archimed number which is the ratio of buoyancy to inertia forces can be expressed below. Ar = g3A60 (1) This nondimensional number will be discussed and compare with the other results later on. In Chapter 3, laminar and turbulence flow are explaning for incompressible fluids. First of all, it was looked the governing equations both for laminar and turbulence cases. If we quik look the equations below, we can understand the other part of study. Laminar Flow Case (governing equations-cartesian coordinates) Mass Conservation 1- - = 0 dx dy (2) Momentum Equations: du du dxx ÖP \dt ox dy) dx rd2xx ] d2^.dx2+dy2) (3) (8w 8v d\] ÖP (d2v dV - + u - + v - = + u - r + - r Ut dx dy) dx \dx2 dy2) (4) Energy Equation dl dl 8T + u - + v- ^2 a2T azT 2 tA +. dt dx dyj p cp v d x dy J (5) Turbulence Flow Case (governing equations-cartesian coordinates) Mass Conservation : p - * - - = 0 dx: (6) Momentum Equation : p au± cm at Jax; ?iJ dx.. au dx. d? 'dx, +P& (7) Energy Equation : P rdi TT di) Kdt 'dx.j (\i dT dX: iV Prax.- -pUiT (8) These equations help us to predict the unknown variable dependent u, v, T, k, e values.For turbulent case there are more equations we need.If we briefly explain these equations we must write them. xiv In order to solve enclosed problem for turbulance the common method from Boussinessq is called eddy viscosity aproximation. According to this method Reynold stress are written below: -UjUj = v, aiL dV \ J-+ dx; dXi 3,J (9) In this equation turbulent viscosity, v, and turbulent Prandtl number at are properties of flow, not fluid. Modelling turbulence is widely investigated subject especially in the last two decades, as a result of rapid developmants in computers and numerical methods. One of these models, called "k-e" model, is very comman, and also forms the fundemantals of the model developed in this thesis. In Chapter 3, fundemantals of k-e model are explained. In this method, the following governing equations are used : dk TT Ök Ö fv, dk] - + Uj = - dt dx; oXj ^ak d x.J + vt fau, sUjI dxs ?+? OX, au; 6X; -8 (10) de ds ~dt+ 'axT vt 5 s dxAGa dx,. + v, 1 + ? ox, dxj ?^-^T-^ıT Ol) ox, k k and solved. The reason for accepting turbulent kinetic energy as the velocity scale and the definition of turbulent kinetic energy is given. Two additional transport equations for turbulent kinetic energy k, and the dissipation of turbulent kinetic energy are solved together with the governing equations. These equations and emprical constants and coefficients are presented in the thesis. Then the insufficiency of this model in flows with low Reynolds numbers are described. Since the analytical solution is imposible, the difficulties for solving k-e turbulent model should be overcome. For that reason modified models developed by various researchers. These developed models are called 'low Reynolds number models" For this model, the differantial equations for turbulent kinetic energy and the dissipation of turbulant kinetic energy are shown below. Dk Dt i a PaXj + "? ak OX: + u.au. P 5XJ f au; au^ L + i ax= ax; -e + D (12) Ds 1 a Dt paxj + fiCel e u,, aUj k p ax= au} au^ -+ IdXj dxj k (13) v, = f"c" - (14) xv Numerical solution consists of discretization by finite difference method and an iterative solution by the means of the code developed in this study. The general form of the dicretized differential equations, variable dependent and independent sources, the flow diagram of the code and the grid distribution are present in the Chapter 4. Grids are generated in such a way that dimensions of the control volumes are very tiny near the walls and they become coarser towords the core region. The grid field is a staggered one that means vectorial dependent variables have been defined at the grid points. In the code, the Power Law Differencing Scheme Method, PLDS, for the convection terms and Semi - Implicit Pressure Linkage Method, SIMPLE, for the pressure correction have been used. Both of these methods are discribed briefly. Also, underrelaxation equations and factors for every variable, and the convergence criteria are the subjects discussed. General boundry conditions are given in Chapter 5. At the walls, velocity, pressure, and turbulent kinetic energy equal to zero; dissipation of kinetic energy, temperature at the horizontol walls are all zero gradiends, while the temperature values at the vertical walls are taken as constant. At inlet, velocity, temperature variables are constant, turbulent kinetic energy is a function of inlet velocity and dissipation of turbulant kinetic energy is a function of the turbulant kinetic at the inlet, while the pressure gradient equals to zero. At outlet section all the dependent variables, except pressure, equal to constant fluxes while the pressure has zero gradient. The results we get, compared between the nondimensional numbers Ar, Re ve Fr. Fr = - 7 r- (Froud number) (15) gs T. V T. oda- US Re = - (Reynolds number) (16) Firstly, when increasing Re, the seperation distance reducing. This means that, since Re number is a charecteristics of turbulence, that result is obvious. Similarly, as the Ar number is increasing, the seperation distance is reducing. It means, the effect of buoyancy into turbulence is very strong. In addition, as the Fr decreasing, seperation distance decreasing as well. This result, completely opposite with the second result.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1998

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1998

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1998

##### Anahtar kelimeler

Sayısal modelleme,
Soğuk hava jeti,
Numerical modelling,
Cold air jet