Contents. 1 Introduction. 2 Analysis

Transcription

1 Contents 1 Introduction Introduction Theory Boundary Conditions Specified Temperature Convection Radiation Applied Heat Flux Method of Solution Analysis Introduction Steady State Analysis Transient Analysis Radiation View Factor Calculation Thermo-Electric Coupling Loads and Boundary Conditions Prescribed Temperature Convection Radiation Applied Heat Flux Heat Generation Time and Temperature Curves Thermal Stress Analysis Thermal Bonding Bonding of Meshes with Noncompatible Elements COSMOSM Advanced Modules i

2 Contents Examples of Bond Connections Guidelines for Using the Bond Capability Phase Change Thermostat Description of Elements Introduction Brief Description of Commands Command Summary Material Properties Loads and Boundary Conditions Time and Temperature Curves Thermal Stress Analysis Thermal Bonding Thermal Analysis Options Postprocessing Commands Likely to be Used for a Given Analysis Steady State Analysis Transient Analysis Detailed Examples Introduction Temperature Distribution on a Plate Given GEOSTAR Input Results An Example of Thermal Bonding Given GEOSTAR Input Results Listing of Session File ii COSMOSM Advanced Modules

3 Part 1 HSTAR Heat Transfer Analysis 6 Verification Problems Index Introduction COSMOSM Advanced Modules iii

4 Contents iv COSMOSM Advanced Modules

5 1 Introduction Introduction The transport of heat can occur through the following modes. Conduction: Thermal energy is transported from one point in a medium to another point through the interaction between the atoms or molecules of the matter. No bulk motion of the matter is involved. Convection: Thermal energy is transported by the moving fluid. Fluid particles act as carriers of thermal energy. Radiation: Thermal energy is transported by electromagnetic waves. No medium is necessary for this type of heat transfer. Our main interest is to consider the conduction heat transfer with the effects of convection and radiation appearing as boundary conditions. Theory The governing equation for conduction heat transfer is as follows. (EQ 1-1) COSMOSM Advanced Modules 1-1

6 Chapter 1 Introduction where, T = Temperature t = Time ρ = Density C = Specific heat Q = Volumetric heat generation rate k x, k y, k z = Thermal conductivities in global X, Y and Z directions, respectively Boundary Conditions Following boundary conditions can be associated with the heat conduction equation. Specified Temperature Temperature can be prescribed on a part of, or on the whole, boundary of the finite element domain. Convection Heat flux = q = h c (T - T ) (EQ 1-2) h c = Heat transfer coefficient T = Surface temperature T = Ambient temperature Radiation Heat flux = q = σ ε (T 4 - T 4 ) (EQ 1-3) σ = Stefan-Boltzmann constant ε = Emissivity 1-2 COSMOSM Advanced Modules

7 Part 1 HSTAR Heat Transfer Analysis T T = Surface temperature = Ambient temperature Applied Heat Flux q = Applied heat flux = - K (EQ 1-4) K = Thermal conductivity = Normal temperature gradient Method of Solution The governing equation along with the specified boundary conditions can be solved using various solution methods. Some of the solution methods commonly used are finite difference and finite element method. Finite element method is more popular because of its ability to handle complicated geometry and the ease with which boundary conditions can be implemented. HSTAR program is based on finite element method. COSMOSM Advanced Modules 1-3

8 1-4 COSMOSM Advanced Modules

9 2 Analysis Introduction The following types of analysis can be performed using HSTAR. Steady state Transient Steady State Analysis Steady state implies that temperature at any given point in the medium is constant with time. In the steady state analysis, the only material property that is needed is the thermal conductivity. Transient Analysis Transient analysis implies that temperature at any given point in the medium varies with time. In the transient analysis, in addition to thermal conductivity, we also need to specify density and specific heat of the material. Whether we consider steady state or transient analysis, nonlinearity comes into picture, when any one of the following conditions is encountered. Temperature dependent material properties Temperature dependent convection coefficient COSMOSM Advanced Modules 2-1

10 Chapter 2 Analysis Temperature dependent heat generation rate Radiation boundary condition Radiation View Factor Calculation The Heat Transfer module (HSTAR) has the capability to perform Radiation View factor calculation for 2D, 3D, and Axisymmetric models. The process requires the definition of a set of radiation source entities along with a pattern of target entities. It is also possible to specify a pattern of blocking entities. Blocking geometric entities stand between the source and target entities and reduce the view factor. The view factors are calculated between each element associated with the source entity and each element associated with the pattern of target entities. If blocking is to be considered, it is necessary to first define the set involving the source and target entities with the blocking option activated. Next a pattern of blocking entities is specified independently. For 2D and Axisymmetric models, the target and blocking entities must be curves, while for 3D models, they can be surfaces or regions. For more details, refer to the RVFTYP and RVFDEF (Analysis > Heat_Transfer > RVF Entity Type and RVF Source/Target) commands. An adaptive view factor calculation option has also been implemented for 3D models. The program will calculate the view factor starting from 4 divisions for each radiation element, and will continue to increase the number of divisions until the computed error is within the user specified tolerance or the number of divisions reaches the maximum allowed which is (20). It is noted that the adaptive calculation method basically corresponds to the adjustment in the number of divisions required for numerical integration. Refer to the RVFTYP (Analysis > Heat_Transfer > RVF Entity Type) command for details. Thermo-Electric Coupling The electric current flow in a conducting medium can produce a considerable amount of heat and this effect is known as Joule heating. HSTAR considers the coupling of the electrical and thermal conduction in which the heat generated due to the current flow along with other specified boundary conditions is used to calculate the temperature distribution. When thermo-electric coupling is considered, we also need to specify the electrical conductivity of the material. At present, only steady state analysis is available. 2-2 COSMOSM Advanced Modules

11 Part 1 HSTAR Heat Transfer Analysis Loads and Boundary Conditions The following thermal boundary conditions and loads are considered in HSTAR. Prescribed Temperature Temperature on a part or whole of the boundary of the model is specified. Convection When a solid is thermally interacting with its surrounding fluid, heat transfer takes place through the convection process in which the motion of the surrounding fluid contributes to the thermal exchange between the solid and the fluid. The boundary condition is applied by specifying the heat transfer coefficient and the ambient temperature of the surrounding fluid. Radiation Generally, heat transfer by radiation becomes significant at high temperatures. The analysis handles radiation between a surface and ambient atmosphere. The user may also specify radiation exchange between bodies. Applied Heat Flux Heat flux entering a surface can be prescribed as a boundary condition. This is equivalent to specifying temperature gradient at the surface. Heat Generation Whereas the above four boundary conditions are applied to a surface heat generation is applied within the material. Joule heating (in which heat is generated within the material due to the resistance to current flow) is an example of heat generation. Heat generation can be prescribed at a node or in an element. Time and Temperature Curves Time curves are used to specify the variation of thermal loads and boundary conditions as function of time. All the thermal boundary conditions and loads discussed above can vary with time and this variation is specified by defining a time curve and associating this curve with the corresponding boundary condition or load. Temperature curves are used to specify the variation of material properties with temperature and they are also used to prescribe the variation of heat transfer coefficient, heat generation rate and surface emissivity with temperature. COSMOSM Advanced Modules

12 Chapter 2 Analysis Thermal Stress Analysis Once a thermal analysis is completed, resulting temperature distribution can be used to calculate thermal stresses in the material. It is now possible to transfer temperature results from transient analysis solution steps as thermal loading to static analysis (up to a maximum of 50 steps). Thermal Bonding The thermal bonding feature allows the user to connect finite element meshes without having to preserve the element type compatibility or mesh continuity at the interface. The geometric entities and corresponding element groups that can be bonded together are shown in the Table 2-1. Table 2-1. Geometric Connections for Using Bond Primary Entity Secondary Entity To Connect... Example CR CR PLANE2D to PLANE2D, or SHELL to SHELL Primary or secondary Primary or secondary Bonding interface CR SF or RG SHELL to SHELL, or SHELL to SOLID Secondary Bonding interface Primary SF or RG SF or RG SOLID to SOLID Primary or secondary Primary or secondary Bonding interface 2-4 COSMOSM Advanced Modules

13 Part 1 HSTAR Heat Transfer Analysis Bonding of Meshes with Noncompatible Elements The bond feature allows the user to connect finite element meshes between any two intersecting geometries without having to preserve the element type compatibility or mesh continuity at the interface. The geometric entities and corresponding element groups that can be bonded together are shown in Table 2-1. In the above table, SHELL refers to all 3-node triangular and 4- or 9-node quadrilateral shell elements that are supported in COSMOSM. Similarly, SOLID refers to 8- or 20-node hexahedral solid elements as well as 4- or 10-node TETRA and 4-node TETRA4R solid elements. Some of the typical applications of the bond command are also shown in the above table. The bond feature is currently applicable to linear static, nonlinear structural, and heat transfer analyses only. The bond capability is specified using the BONDING submenu from LoadsBC > STRUCTURAL. The BONDDEF (LoadsBC > STRUCTURAL > BONDING > Define Bond Parameter) command bonds faces of elements associated with the selected geometric entities. The user specifies a primary bond entity (curve, surface, or region) and a pattern of target entities (curves, surfaces, or regions). All geometric entities must have meshing completed before issuing this command in order to generate the bond information. Element edges/faces associated with the primary geometric entity are bonded with edges/faces of the secondary entities. The command is useful in connecting parts with incompatible mesh at the interface. The BONDLIST (LoadsBC > STRUCTURAL > BONDING > List) command can be used to list a pattern of bond sets previously defined by the BONDDEF (LoadsBC > STRUCTURAL > BONDING > Define Bond Parameter) command. A typical listing is as follows: Stype Source Ttype #Target Targets CR 53 SF 1 7 CR 50 SF 1 7 CR 47 SF 1 7 CR 44 SF 1 7 CR 41 SF 1 7 COSMOSM Advanced Modules

14 Chapter 2 Analysis Examples of Bond Connections The following figures show examples of non-compatible connections where bonding is useful. Figure 2-1a. Solid-to-Shell Connection Figure 2-1b. Shell-to-Shell Connection Figure 2-1c. Solid-to-Solid Connection 2-6 COSMOSM Advanced Modules

15 Part 1 HSTAR Heat Transfer Analysis Figure 2-1d. Shell-to-Shell Connection Figure 2-1e. Shell-to-Shell Connection Guidelines for Using the Bond Capability The following points should be considered in the application of this command: The BONDDEF (LoadsBC > STRUCTURAL > BONDING > Define Bond Parameter) command internally uses constraint equations to match the displacements and rotations of the two parts. The quality assurance tests have shown that for parts with reasonable stiffness properties and mesh densities, the maximum displacement and stress values obtained from the bond command are within ten percent of those values obtained from a merged model with compatible elements and coincident nodes. COSMOSM Advanced Modules

16 Chapter 2 Analysis The above command is currently applicable to linear static analysis, buckling, natural frequency computations, heat transfer analysis, and nonlinear structural analysis only. Figure 2-2. Uni-directional and Bi-directional Bonding Unidirectional For both types of bonding: Primary or the source entity is always the one that has fewer degrees of freedom Secondary or the target entity is always the one that has larger number of degrees of Source Target Source or Target Target or Source (Same Element Type) (Same Element Type) Source Target Source Target (Same Element Type) (Same Element Type) Bidirectional Source Target Source or Target Target or Source (Same Element Type) (Same Element Type) 2-8 COSMOSM Advanced Modules

17 Part 1 HSTAR Heat Transfer Analysis The BONDDEF (LoadsBC > STRUCTURAL > BONDING > Define Bond Parameter) command offers the option of choosing between uni-directional bond (i.e. connecting all the nodes on primary entity to the elements on the secondary entity) or bi-directional bond (i.e. connecting the nodes on each entity to the elements on the other entity). The one directional bond should be used when connecting lower order elements of the primary (source) entity to lower or higher order elements of the secondary (target) entity. The bi-directional bond should be used in connecting higher order elements of the primary entity to higher order elements of the secondary entity. The following figure illustrates uni-directional and bi-directional bonding. When bonding solids and shells, it is advisable to use shells as the source and solids as the target irrespective of the element order. When shell elements are connected to solid elements, the common nodes at the boundary should not be merged as this will free the rational degrees of shell at that node. Actually, it is advantageous not to have coincident nodes at all in such problems. In shell to shell, or, solid to solid connections, merging of the coincident nodes at the boundary is allowed. In problems where the stress concentration at the bonded intersection is critical, both parts should have a fine mesh in this region, even though the two meshes are not matching (see figure below). You may first perform an analysis with coarse mesh to determine the area requiring fine mesh. Figure 2-3 Replace the gap by a surface or a region type entity and fill with a fine mesh. Top plate Bottom plate Bonding surface. Use fine mesh in this area based on results from a coarse one. Bonding curves COSMOSM Advanced Modules

18 Chapter 2 Analysis The results obtained from the BONDDEF (LoadsBC > STRUCTURAL > BONDING > Define Bond Parameter) command may deteriorate in problems where a rigid part is connected to a relatively flexible part. The bonded area in the flexible part undergoes warping or has high displacement gradients. The results will improve if the mesh density for the flexible part is increased in the bonded area. The actual constraint relations between the nodes of source and target geometric entities are formed and computed in the analysis stage. Phase Change When a material changes its phase from/to solid, liquid, or gas, it either generates or absorbs heat. The heat associated with phase change is called latent heat. HSTAR supports phase change by associating the material property enthalpy with a temperature curve, with a sudden rise or drop at the temperature of phase change. HSTAR uses this information to calculate and use the latent heat absorbed or generated by the material. Thermostat In transient studies, you can control heat power and heat flux conditions by defining a thermostat. The thermostat is defined by a sensor location (node), a temperature range (cut-in and cut-out temperatures), and a temperature curve to determine the associated heat generation/dissipation boundary conditions. The thermostat is considered a heater if the cut-in temperature is lower than the cutout temperature and a cooler if the cut-in temperature is higher than the cut-out temperature regardless of the associated boundary conditions. Before starting a solution step, the program checks the temperature of the sensor. If the thermostat is a heater, the thermostat is turned on during the next solution step if the temperature of the node at the sensor is lower than the cut-out temperature and the device is generating heat. If the thermostat is a cooler, the thermostat is turned on during the next solution step if the temperature of the node at the sensor is higher than the cut-out temperature and the device is dissipating heat. Refer to the THSTAT (LoadsBC, THERMAL, THERMOSTAT, Define) command for details COSMOSM Advanced Modules

19 3 Description of Elements Introduction The table on the next page lists the elements supported by the HSTAR module. COSMOSM Advanced Modules 3-1

20 Chapter 3 Description of Elements Table 3-1. Elements for Thermal Analysis (HSTAR) Element Type 2D Spar/Truss 2D Elastic Beam 3D Elastic Beam 3D Spar/Truss General Mass Element Radiation Link Convection Link 2D 4- to 8-node Plane Stress, Strain, Body of Revolution 3D 3- to 6-node Plane Stress, Strain, Body of Revolution Triangular Thick Shell Quadrilateral Thick Shell 6-Node Triangular Thin Shell 6-Node Triangular Thick Shell 3D 8- to 20-node Continuum Brick 8-node Composite Solid 3D 4-node Tetrahedron Solid 3D 4-node Tetrahedron Solid with Rotation 3D 10-node Tetrahedron Solid Quadrilateral Composite Plate and Shell Triangular Thin Shell Quadrilateral Thin Shell 4-node Hydraulic Link Element Thermal-Fluid Element Element Name TRUSS2D BEAM2D BEAM3D TRUSS3D MASS RLINK CLINK PLANE2D TRIANG SHELL3T SHELL4T SHELL6 SHELL6T SOLID SOLIDL TETRA4 TETRA4R TETRA10 SHELL4L SHELL3 SHELL4 HLINK FLUIDT We can also broadly categorize the elements based on the dimensionality of the problem. TRUSS2D, TRUSS3D, BEAM2D, and BEAM3D elements are used for one dimensional analysis. PLANE2D, TRIANG, SHELL3T, SHELL4T, SHELL3, SHELL4, and HLINK are used for two dimensional problems. SOLID, SOLIDL, TETRA4 and TETRA10 are used for three dimensional problems. CLINK and RLINK elements could be used for any type of problem. SHELL4L is used for analyzing layered composite materials. For a detailed description of all the above elements, refer to the Element Library chapter in the COSMOSM User s Guide. 3-2 COSMOSM Advanced Modules

21 Chapter 3 Description of Elements COSMOSM Advanced Modules 3-3

22 Chapter 3 Description of Elements 3-4 COSMOSM Advanced Modules

23 4 Brief Description of Commands Command Summary Solving a typical thermal problem using finite element method involves generating a proper finite element mesh, imposing initial and boundary conditions and running the analysis. The following sections give a brief description of commands that are used in prescribing boundary conditions, specifying analysis options and solution parameters. Commands used for a typical 2D analysis are described and similar commands are available for 3D analysis. Material Properties For a steady state analysis we need only to specify thermal conductivity and for a transient analysis, in addition to thermal conductivity we also need to define density and specific heat. For thermo-electric coupling, it is also necessary to define the value of electrical conductivity. All the material properties are defined using MPROP (Propsets > Material Property) command from the Propsets menu. Loads and Boundary Conditions Nodal temperatures at individual nodes and all nodes associated with a curve, contour, region, surface and volume are defined using the LoadsBC > THERMAL > TEMPERATURE menu. Convection film coefficients and the associated ambient temperatures are specified using the LoadsBC > THERMAL > CONVECTION COSMOSM Advanced Modules 4-1

24 Chapter 4 Brief Description of Commands submenu. Radiation energy exchange between a surface and the ambient atmosphere is specified using the LoadsBC > THERMAL > RADIATION menu. Heat flux entering or leaving a surface can be prescribed using LoadsBC > THERMAL > HEAT FLUX menu. Heat generation can be specified at point or volumetric sources. Nodal heat generation is specified using the LoadsBC > THERMAL > NODAL HEAT menu. Element heat generation is specified using the LoadsBC > THERMAL > ELEMENT HEAT menu. For modeling heat transfer due to flow in a pipe, the HLINK element can be used and the input for thermal boundary conditions is specified using LoadsBC > THERMAL > HYDRAULIC FLOW menu. For radiation heat exchange between multiple bodies, the view factors are automatically calculated by the program using the following commands from Analysis > HEAT TRANSFER menu: RVF Entity Type (RVFTYP), RVF Source/ Target (RVFDEF), Del Rad View Factor (RVFDEL) and List Rad View Factor (RVFLIST). Time and Temperature Curves Time curves are used to specify the variation of thermal loads and boundary conditions as function of time. All the thermal boundary conditions and loads can vary with time. Temperature curves are used to specify the variation of material properties with temperature and they are also used to prescribe the variation of heat transfer coefficient and heat generation rate with temperature. Using a time or temperature curve involves the following steps. Define time or temperature curve using the CURDEF (LoadsBC > FUNCTION CURVE > Time/Temp Curve) command. This curve is automatically activated. Define the entity of interest (boundary condition, load, material property, etc.). Deactivate the curve using ACTSET (Control > ACTIVATE > Set Entity) command so that this curve is not inadvertently associated with some other entity defined later on. For example, we want to prescribe a time varying temperature boundary condition. First issue CURDEF (LoadsBC > FUNCTION CURVE > Time/Temp Curve) command to define time curve. Next, define the nodal temperature at the beginning of the curve. Deactivate the curve association after you have finished the timedependent input. 4-2 COSMOSM Advanced Modules

25 Part 1 HSTAR Heat Transfer Analysis Geo Panel: LoadsBC > THERMAL > TEMPERATURE > Define Nodes (NTND) Define nodal temperatures... Geo Panel: Control > ACTIVATE > Set Entity (ACTSET) Set label > TC: Time Curve Time curve label > 0 An example of the use of a temperature curve for prescribing a material property variation is (after defining the temperature curve): Geo Panel: Propsets > Material Property (MPROP) Define thermal conductivity (kx)... Geo Panel: Control > ACTIVATE > Set Entity (ACTSET) Set label > TC: Temperature Curve Temperature curve label > 0 Thermal Stress Analysis Once a thermal analysis is completed, resulting temperature distribution can be used to calculate thermal stresses in the material. The following steps can be used to calculate thermal stresses. Complete the thermal analysis. Use TEMPREAD (LoadsBC > LOAD OPTIONS > Read Temp as Load) command to specify the time step at which thermal stress analysis is to be done. Activate the thermal loading using the A_STATIC (Analysis > STATIC > Static Analysis Options) command. Run the static analysis using R_STATIC (Analysis > STATIC > Run Static Analysis) command. Thermal Bonding The bonding feature can be used to handle problems in which adjacent geometric entities (as curves, surfaces or regions or combinations of these) are meshed in an incompatible manner. The BONDDEF (LoadsBC > STRUCTURAL > BONDING > Define Bond Parameter) command is used to specify the interfaces along which mesh incompatibility exists. COSMOSM Advanced Modules 4-3

26 Chapter 4 Brief Description of Commands Thermal Analysis Options HSTAR is capable of solving both steady state and transient problems. The type of analysis (steady state or transient) is set using the A_THERMAL (Analysis > HEAT TRANSFER > Thermal Analysis Options) command. By default, steady state analysis is performed. A_THERMAL (Analysis > HEAT TRANSFER> Thermal Analysis Options) command also specifies convergence parameters for nonlinear problems and analysis options for thermo-electric coupling. For transient problems, the total solution time and time step are prescribed using the TIMES (LoadsBC > LOAD OPTIONS > Time Parameter) command. Initial distribution of temperature is input by the INITIAL (LoadsBC > LOAD OPTIONS > Initial Cond) command. The printing and plotting of output results from a transient analysis is controlled by the HT_OUT (Analysis > HEAT TRANSFER > Thermal Output Options) command. Postprocessing The output generated by the thermal analysis can be viewed graphically in GEOSTAR. Issue the Results > PLOT > Thermal command to load temperature, gradient or heat flux values into memory and plot the loaded data. We can also look at the time history of temperature, gradient, etc. at any node. First issue the ACTXYPOST (Display > XY PLOTS > Activate Post-Proc) to load proper data into memory and then issue XYPLOT (Display > XY PLOTS > Plot Curves) to plot the time history. Commands Likely to be Used for a Given Analysis The following section gives a brief description of commands that may be necessary to run a given type of analysis once a proper finite element mesh is generated. This is intended as a general guideline only because the problem at hand may not need all the commands that are mentioned below or it may need some other commands which are not mentioned. The commands are given for a typical 2D problem and similar commands are available for 3D problems. 4-4 COSMOSM Advanced Modules

27 Part 1 HSTAR Heat Transfer Analysis Steady State Analysis Command (Cryptic) MPROP (Propsets > Material Property) RCONST (Propsets > Real Constant) NTCR (LoadsBC > THERMAL > TEMPERATURE > Define Curves) CECR (LoadsBC > THERMAL > CONVECTION > Define Curves) QESF (LoadsBC > THERMAL > ELEMENT HEAT > Define Surfaces) QSF (LoadsBC > THERMAL > NODAL HEAT > Define Surfaces) HFND (LoadsBC > THERMAL > HEAT FLUX > Define Nodes) NPRND (LoadsBC > FLUID FLOW > PRESSURE > Define Nodes) HXCR (LoadsBC > THERMAL > HEAT FLUX > Define Curves) RECR (LoadsBC > THERMAL > RADIATION > Define Curves) RVFTYP (Analysis > HEAT TRANSFER > RVF Entity Type) RVFDEF (Analysis > HEAT TRANSFER > RVF Source/Target) Intended Use Specify material properties Specify real constants Specify nodal temperature boundary conditions Specify convection boundary conditions Specify element heat generation rate Specify nodal heat generation rate Specify nodal fluid flow rate (for HLINK element) Specify nodal pressure (for HLINK element) Specify heat flux boundary condition Specify radiation boundary condition Specify analysis options for thermal radiation exchange Specify radiation exchange between bodies COSMOSM Advanced Modules 4-5

28 Chapter 4 Brief Description of Commands Command (Cryptic) CURDEF (LoadsBC > FUNCTION CURVE > Time/Temp Curve) BONDDEF (LoadsBC > STRUCTURAL > BONDING > Define Bond Parameter) A_THERMAL (Analysis > HEAT TRANSFER > Thermal Analysis Options) R_THERMAL (Analysis > HEAT TRANSFER > Run Thermal Analysis) Intended Use Specify temperature curve for defining temperature dependent material properties Define bonding at interfaces of geometric entities which are meshed in an incompatible manner Specify thermal analysis options Run the analysis Transient Analysis In addition to the above commands for a steady state problem, it is necessary to issue the following commands for a transient problem. Command (Cryptic) CURDEF (LoadsBC > FUNCTION CURVE > Time/Temp Curve) TIMES (LoadsBC > LOAD OPTIONS > Time Parameter) INITIAL (LoadsBC > LOAD OPTIONS > Initial Cond) HT_OUTPUT (Analysis > HEAT TRANSFER > Thermal Output Options) Intended Use Define a time curve which specifies the time variation of loads and boundary conditions Specify the total solution time and time step Specify the initial temperature distribution Specify printing and plotting intervals for the results from thermal analysis 4-6 COSMOSM Advanced Modules

29 5 Detailed Examples Introduction This example is a typical heat transfer analysis problem solved by the HSTAR module of COSMOSM through GEOSTAR. A detailed description of the required steps to set up and solve the problem is furnished. Temperature Distribution on a Plate Determine the temperature distribution in a plate subjected to temperature and convection boundary conditions. Consider the effect of constant heat generation on the plate. The plate is shown in Figure 5-1. Figure 5-1 Y Z B C T = 100 F a h = BTU / in 2 - sec - F h A a D T = 10 F X COSMOSM Advanced Modules 5-1

30 Chapter 5 Detailed Examples Given Thickness of plate = h = 1 in Side of plate = a = 10 in Temperature on edge AB = 100 F Ambient temperature = 10 F Thermal conductivity of steel = BTU/in sec F Constant heat generation = BTU/in 3 sec Convective heat transfer coefficient on the edge DC = BTU/in 2 sec F GEOSTAR Input Input the problem step-by-step with GEOSTAR commands and perform thermal analysis. Node generation commands will not be discussed in detail. 1. Define the element group. For this example, the 2D plane stress element is selected. Geo Panel: Propsets > Element Group (EGROUP) Element group > 1 Element category > Area Element type (for area) > PLANE2D Accept defaults Define the Thickness of the Plane Stress element. Geo Panel: Propsets > Real Constant (RCONST) Associated element group > 1 Real constant set > 1 Start location of the real constants > 1 No. of real constants to be entered > 2 RC1: Thickness > 1 RC2: Material angle (Beta) > COSMOSM Advanced Modules

31 Part 1 HSTAR Heat Transfer Analysis 3. Define thermal conductivity. Geo Panel: Propsets > Material Property (MPROP) Material property set > 1 Material property name > kx Property value > Since the material is isotropic, this thermal conductivity value is used by default in all directions, i.e., K x = K y = K z. 4. The geometry of the model is created next. Change the view to X-Y using the viewing icon. Define the X-Y plane on which the surface is created as follows: Define the xy plane. Geo Panel: Geometry > GRID > Plane (PLANE) Rotation/sweep axis > Z Offset on axis > 0.0 Grid line style > Solid Geo Panel: Geometry > SURFACES > Define w/4 Coord (SF4CORD) Surfaces > 1 XYZ-coordinate value of Keypoint 1> 0,0,0 XYZ-coordinate value of keypoint 2 > 10,0,0 XYZ-coordinate value of keypoint 3 > 10,10,0 XYZ-coordinate value of keypoint 4 > 0,10,0 5. Define elements and nodes through mesh generation. Geo Panel: Meshing > PARAMETRIC MESH > Surfaces (M_SF) Beginning surface > 1 Ending surface > 1 Increment > 1 Number of nodes per element > 4 Number of elements on 1st curve > 5 Number of elements on 2nd curve > 5 Spacing ratio for 1st curve > 1.0 Spacing ratio for 2nd curve > 1.0 COSMOSM Advanced Modules 5-3

32 Chapter 5 Detailed Examples 6. See the Auto scale icon to properly view the model. Define temperature boundary conditions along curve 3. Geo Panel: LoadsBC > THERMAL > TEMPERATURE > Define Curves (NTCR) Beginning curve > 3 Value > 100 Ending curve > 3 Increment > 1 7. Define convection boundary conditions along curve 4. Geo Panel: LoadsBC > THERMAL > CONVECTION > Define Curves (CECR) Beginning curve > 4 Convection coefficient > Ambient temperature > 10 Ending curve > 4 Increment > 1 Time curve for ambient temperature > 0 8. The constant heat generation rate is specified using the QESF (LoadsBC > THERMAL > ELEMENT HEAT > Define Surfaces) command. Geo Panel: LoadsBC > THERMAL > ELEMENT HEAT > Define Surfaces (QESF) Beginning surface > 1 Value > Ending surface > 1 Increment > 1 9. The thermal analysis option by default is steady state thus the A_THERMAL (Analysis > HEAT TRANSFER > Thermal Analysis Options) command is not required. Just use the R_THERMAL (Analysis > HEAT TRANSFER > Run Thermal Analysis) command to run the heat transfer program. When the analysis is completed, the program will return to GEOSTAR. Next use the EDIT (FILE > Edit a File) command or your favorite editor to view the output file (*.TEM). Use the ACTTEMP and TEMPPLOT (Results > PLOT > Thermal) commands to generate a temperature contour plot. 5-4 COSMOSM Advanced Modules

33 Part 1 HSTAR Heat Transfer Analysis Results Temperature at node 24: Analytical solution HSTAR solution = F = F An Example of Thermal Bonding The following example illustrates the use of the BONDING feature in thermal analysis. The problem is to find the temperature distribution in a plate which is subjected to temperature boundary conditions. To illustrate the bonding capability of the HSTAR program, the plate is divided into two surfaces which are meshed in such a way that the meshing is incompatible at the interface of the two surfaces. Figure 5-2 Y Node 49 B C b T = 0 C T = 100 C A D X Given Thickness of the plate = 1 cm Length of the plate = l = 2 m Width of the plate = 1 m Temperature on edge AB = 0 C Temperature on edge CD = 100 C Thermal conductivity of the material = 1 W/m - K COSMOSM Advanced Modules 5-5

34 Chapter 5 Detailed Examples GEOSTAR Input The following is a step by step procedure to generate the required input and perform the thermal analysis. 1. Define the element group 2D Plane stress element is selected. Geo Panel: Propsets > Element Group (EGROUP) Element group > 1 Element category > Area Element type (for area) > PLANE2D Accept defaults Define the thickness of the Plane stress element through a real constant set. Geo Panel: Propsets > Real Constant (RCONST) Associated element group > 1 Real constant set > 1 Start location of the real constants > 1 No. of real constants to be entered > 2 RC1: Thickness > 0.01 RC2: Material angle (Beta) > Define thermal conductivity. Geo Panel: Propsets > Material Property (MPROP) Material property set > 1 Material property name > kx Property value > Define the geometry of the model. Change the view to X-Y using the Viewing icon. Define the X-Y plane on which the surface is created as follows: Geo Panel: Geometry > GRID > Plane (PLANE) Rotation/sweep axis > Z Offset on axis > 0.0 Grid line style > Solid 5-6 COSMOSM Advanced Modules

35 Part 1 HSTAR Heat Transfer Analysis Geo Panel: Geometry > SURFACES > Define w/4 Coord (SF4CORD) Surface > 1 XYZ-coordinate of keypoint 1 > 0,0,0 XYZ-coordinate of keypoint 2 > 1,0,0 XYZ-coordinate of keypoint 3 > 1,1,0 XYZ-coordinate of keypoint 4 > 0,1,0 Generate an additional surface by translating the first surface in the x-direction by 1 m. Geo Panel: Geometry > SURFACES > GENERATION MENU > Generate (SFGEN) Generation number > 1 Beginning surface > 1 Ending surface > 1 Increment > 1 Generation flag > Translation X-displacement > 1.0 Y-displacement > 0.0 Z-displacement > Use the Auto scale option to see the model clearly. Define elements and nodes through mesh generation. Note that the two surfaces are meshed separately to create incompatibility at the interface of the two surfaces. Geo Panel: Meshing > PARAMETRIC MESH > Surfaces (M_SF) Beginning surface > 1 Ending surface > 1 Increment > 1 Number of nodes per element > 4 Number of elements on 1st curve > 5 Number of elements on 2nd curve > 5 Accept defaults... Geo Panel: Meshing > PARAMETRIC MESH > Surfaces (M_SF) Beginning surface > 2 Ending surface > 2 Increment > 1 Number of nodes per element > 4 COSMOSM Advanced Modules 5-7

36 Chapter 5 Detailed Examples Number of elements on 1st curve > 5 Number of elements on 2nd curve > 4 Accept defaults Merge the coincident nodes Geo Panel: Meshing > NODES > Merge (NMERGE) Accept defaults Define temperature boundary conditions along the left and right edges of the plate. Geo Panel: LoadsBC > THERMAL > TEMPERATURE > Define Curves (NTCR) Beginning curve > 3 Value > 0 Ending curve > 3 Increment > 1 Geo Panel: LoadsBC > THERMAL > TEMPERATURE > Define Curves (NTCR) Beginning curve > 5 Value > 100 Ending curve > 5 Increment > 1 8. Define bonding between the two surfaces Geo Panel: LoadsBC >STRUCTURAL >BONDING >Define Bond Parameter (BONDDEF) Bonding set > 1 Primary geometric entity type > Curve Primary curve > 4 Secondary geometric entity type > Curve Beginning curve > 4 Ending curve > 4 Increment > 1 Direction flag > Bi Dir 5-8 COSMOSM Advanced Modules

37 Part 1 HSTAR Heat Transfer Analysis 9. Run the thermal analysis. Geo Panel: Analysis > HEAT TRANSFER > Run Thermal Analysis (R_THERMAL) After the analysis is completed, the program will return to GEOSTAR. Use the ACTTEMP and TEMPPLOT (Results > PLOT > Thermal) command to generate a temperature plot. Results Temperature at node 49: Analytical solution HSTAR solution (with bonding) HSTAR solution (without bonding) = 50 C = 50 C = 75.9 C Listing of Session File EGROUP,1,PLANE2D,0,1,0,0,0,0,0, RCONST,1,1,1,2,0.01,0, MPROP,1,KX,1.0, PLANE,Z,0,1, VIEW,0,0,1,0, SCALE,0, SF4CORD,1,0,0,0,1,0,0,1,1,0,0,1,0, SFGEN,1,1,1,1,0,1.0,0,0, M_SF,1,1,1,4,5,5,1,1, M_SF,2,2,1,4,5,4,1,1, NMERGE,1,66,1,0.0001,0,1,0, NTCR,3,0.0,3,1, NTCR,5,100.0,5,1, BONDDEF,1,0,4,0,4,4,1,2, R_THERMAL COSMOSM Advanced Modules 5-9

38 5-10 COSMOSM Advanced Modules

39 6 Verification Problems Introduction In the following, a comprehensive set of verification problems are provided to illustrate the various features of the heat transfer analysis module (HSTAR). The problems are carefully selected to cover a wide range of applications in the field of thermal analysis. The input files for the verification problems are available in the...\vprobs\ HeatTransfer folder. Where... denotes the COSMOSM installation folder. For example the input file for problem TL01 is...\vprobs\heattransfer\tl01.geo. COSMOSM Advanced Modules 6-1

40 Linear Heat Transfer Analysis 6-2 COSMOSM Advanced Modules

41 Part 1 HSTAR Heat Transfer Analysis TL01: Steady State Heat Conduction in a Square Plate TYPE: Steady state heat conduction with prescribed temperature boundary conditions, SHELL3T elements are used. REFERENCE: Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids, 2nd edition, Oxford University Press, PROBLEM: Determine the temperature at the center of a square plate with prescribed edge temperatures. GIVEN: Thermal Conductivity = 43 w/m C Boundary Conditions: Along the edge AB, temp. = 0 C Along the edge BC, temp. = 0 C Along the edge CD, temp. = 0 C Along the edge DA, temp. = 100 C Width and Height of Plate = 4 m MODELING HINTS: Since the plate and boundary conditions are symmetrical about cross-section I-I, only one half of the plate is modeled using SHELL3T elements as shown in the figure. ANALYTICAL SOLUTION: Temperature at any point (x,y) in the plate is: COSMOSM Advanced Modules 6-3

42 Chapter 6 Verification Problems Where a = The length of a side of plate T o = The temperature at x = 0 COMPARISON OF RESULTS: At the center of the plate (Node 41). Temperature C Theory 25 COSMOSM 25 Difference 0% Figure TL01-1 Y D 0 C I 0 I A 0 B X Finite Element Model Problem Sketch 6-4 COSMOSM Advanced Modules

43 Part 1 HSTAR Heat Transfer Analysis TL02: Steady State Heat Conduction in an Orthotropic Plate TYPE: Steady state heat conduction with convection boundary conditions, SHELL4 elements are used. REFERENCE: M. N. Ozisik, Heat Conduction, Wiley, New York, PROBLEM: Determine the temperature distribution in an orthotropic plate with a constant rate of heat generation. The boundaries at x = 0 and y = 0 are insulated, and those at x = a and y = b are dissipating heat by convection into the atmosphere which is at zero temperature. GIVEN: Thermal Conductivity along x direction = K x = 10 w/m C along y direction = K y = 20 w/m C Convection Heat Transfer Coefficient at the boundary BC = h 1 = 10 w/m 2 C at the boundary DC = h 2 = 20 w/m 2 C Length of the plate = a = 1 m Width of the plate = b = 2 m Thickness of the plate = 0.1 m Rate of heat generation Q = 100 w/m 3 MODELING HINT: Plate is modeled using 200 SHELL4 elements. ANALYTICAL SOLUTION: COSMOSM Advanced Modules 6-5

44 Chapter 6 Verification Problems Where: K 1 = K K 2 = Ky COMPARISON OF RESULTS: Node X (m) Theory COSMOSM Figure TL02-1 D T = 0 C h = 20 w/m 2 C 2 C y Insulated b T = 0 C h = 10 w/m 2 C 1 Insulated A a Problem Sketch B 1 11 Finite Element Model x 6-6 COSMOSM Advanced Modules

45 Part 1 HSTAR Heat Transfer Analysis TL03: Transient Heat Conduction in a Long Cylinder TYPE: Transient heat conduction with convection boundary conditions, PLANE2D elements. REFERENCE: J. P. Holman, Heat Transfer, McGraw-Hill Book Company, 1976, p PROBLEM: A long aluminum cylinder, 5 cm in diameter and initially at 200 C, is suddenly exposed to a convection environment at 70 C and h = 525 W/m 2 C. Calculate the temperature at a radius of 1.25 cm, one minute after the cylinder is exposed to the environment. GIVEN: Radius of cylinder = r o = m Thermal conductivity = K = 215 W/m C Mass density = ρ = 2700 Kg/m 3 Specific heat = C = J / Kg C Initial temperature = T 0 = 200 C Ambient temperature = T = 70 C Convective heat transfer coefficient = h = 525 w/m 2 C MODELING HINTS: Since the cylinder and boundary conditions are axisymmetric, PLANE2D axisymmetric elements are used to model this problem. COSMOSM Advanced Modules 6-7

46 8h, T Chapter 6 Verification Problems COMPARISON OF RESULTS: Comparison of solutions is made at r = m (node 21) and at t = 60 sec: Temperature C Theory COSMOSM Figure TL03-1 Y r o Y X r o X Finite Element Model Z Problem Sketch 6-8 COSMOSM Advanced Modules

47 Part 1 HSTAR Heat Transfer Analysis TL04: Thermal Stresses in a Hollow Cylinder TYPE: Thermal stress analysis, PLANE2D axisymmetric element. REFERENCE: Timoshenko and Goodier, Theory of Elasticity, McGraw-Hill Book Co., New York, PROBLEM: The hollow cylinder in plane strain is subjected to two independent loading conditions. 1. An internal pressure Pa 2. A steady state axisymmetric temperature distribution due to the following boundary conditions. At r = 1, temperature = 100 F At r = 2, temperature = 0 F Pressure and Temperature Loading PLANE2D Axisymmetric Model. GIVEN: E = 30 x 10 6 psi a = 1 in b = 2 in ν = 0.3 α x = 1 *10-6 / F Kx = 1 Btu/in sec F Pa = 100 psi T a = 100 F T b = 0 F COSMOSM Advanced Modules 6-9

48 Chapter 6 Verification Problems COMPARISON OF RESULTS: Theory COSMOSM Temperature in F Node Node Stress at r=1.325 (Center of Element 7) in psi T r (SX) i i T θ (SZ) i Figure TL04-1 Tθ Pa T r Problem Sketch y x a b C L Finite Element Model 6-10 COSMOSM Advanced Modules

49 Part 1 HSTAR Heat Transfer Analysis TL05: Heat Conduction Due to a Series of Heating Cables TYPE: Steady state heat conduction due to internal heat generation (PLANE2D elements). REFERENCE: J. N. Reddy, An introduction to the finite element method. McGraw-Hill Book Co., 1984, p PROBLEM: A series of heating cables have been placed in a conducting medium as shown in figure. The medium has conductivities of K x = 10 w/cm K and K y = 15 w/cm K. The Upper surface is exposed to a temperature of -5 C, and the lower surface is bounded by an insulating medium. Assuming that each cable is a point source of 250 w, determine the temperature distribution in the medium. GIVEN: Thermal conductivity in: x direction K x = 10 w/cm K y direction K y = 15 w/cm K Ambient temperature T = 268 K Convection coefficient h = 5 w/cm 2 K Rate of heat generation in the cable per unit length Q= 250 w MODELING HINTS: Since the cables are uniformly distributed throughout the medium, the problem can be simplified by analyzing only the section ABCD as shown in the figure. Because of symmetry, consider the sides AD and BC to be insulated. Since the medium is symmetric about x-y plane, plane strain option of PLANE2D elements has been selected. COSMOSM Advanced Modules 6-11

50 Chapter 6 Verification Problems Figure TL05-1 T 8 = 268 K Y h = 5 w/cm 2 K Y Cables D C D C 2 Cabl Insulated A 4 B X A B 1 9 Finite Element Model X Problem Sketch 6-12 COSMOSM Advanced Modules

51 Part 1 HSTAR Heat Transfer Analysis TYPE: Seepage flow, PLANE2D elements. TL06: Pressure Distribution in an Aquifer Flow REFERENCE: J. N. Reddy, An introduction to the finite element method, McGraw-Hill Book Co., 1984, p PROBLEM: A well penetrates an aquifer and pumping is done at a rate of Q = 150 m 3 /h. The permeability of the aquifer is K = 25 m 3 /(hm 2 ). The aquifer is unconfined and radial symmetry exists in the flow field (with the origin of the radial coordinate being at the pump). A constant head of U = 50 m exists at a radial distance of L = 200 m. Determine the distribution of piezometric head. GIVEN: Permeability of aquifer = K = 25 m 3 /(h m 2 ) piezometric head (at r = 200 m) = U = 50 m Rate of pumping = Q = 150 m 3 /h MODELING HINTS: This problem is modeled by PLANE2D elements. Since the distribution of pressure in the radial direction is a function of logarithm of radial coordinate, variable node spacing is used to get better results. The ratio of last division size to the first division size along the radial direction is assumed to be 6. This problem has been solved using two types of PLANE2D elements. Case A Plane strain option of PLANE2D elements has been selected. This type of model is especially useful to visualized piezometric head contours (which are concentric circles). Case B Axial symmetry of the problem is used to simplify the model. Axisymmetric option of PLANE2D elements has been selected. Note that the governing equation of this COSMOSM Advanced Modules 6-13

52 Chapter 6 Verification Problems problem is similar to that of steady state heat conduction in radial direction. Hence this problem has been solved by identifying the variables as shown in Table 6-1 Table TL06-1. Interpretation of Heat Conduction Variables in Seepage Problem Variable Steady State Heat Conduction Pressure Distribution is an Aquifer Flow u Temperature Piezometric head K Thermal conductivity Permeability coefficient Q Internal heat generation Recharge r Radial coordinate Radial coordinate ANALYTICAL SOLUTION: The governing equation for an unconfined aquifer with flow in the radial direction is given by: Where: r = radial coordinate Q = recharge K = coefficient of permeability u = piezometric head Note that pumping is considered to be a negative recharge. The associated boundary conditions are at r = 0 Q = recharge r = L u = u 0 Solution of the above differential equation is given by 6-14 COSMOSM Advanced Modules

53 Part 1 HSTAR Heat Transfer Analysis COMPARISON OF RESULTS: Piezometric head at r = (at Node 5) Theory COSMOSM Case A COSMOSM Case B Head (m) Figure TL06-1 Z C L Y X X L L C L Case A Case B Problem Sketch and Finite Element Model COSMOSM Advanced Modules 6-15

54 Chapter 6 Verification Problems TL07: Potential Flow Over a Cylinder Confined Between Two Walls TYPE: Potential flow: stream function and velocity potential formulations REFERENCE: Irving H. Shames, Mechanics of Fluids, McGraw-Hill Book Co., PROBLEM: Consider an infinitely long cylinder at rest in a large body of fluid flowing uniformly at right angles to the axis of the cylinder. Assuming irrotational and incompressible flow, find the maximum velocity of the flow. Solve the problem using the Stream function formulation. Figure TL07-1 D E V o I H J L K 0.1 m G C d GIVEN: Diameter of cylinder = d = 0.2 m Velocity = V 0 = 1.0 m/s A d F Problem Sketch B MODELING HINTS: This problem has been modeled by PLANE2D elements. Note that the model is symmetric about the axes EG and HF. Hence it is sufficient to analyze one quarter of the model with the appropriate boundary conditions on the axes of symmetry. Assume that the velocity is constant at a distance of 1 m from the axis of cylinder. Since the gradients of stream function are very high near the cylinder, variable mesh spacing has been selected. Note that the variable finite element mesh can be generated very easily using mesh generation commands COSMOSM Advanced Modules

55 Part 1 HSTAR Heat Transfer Analysis Stream Function Formulation The incompressible steady flow may be represented by Laplace equation: For a two dimensional flow, the above equation can be rewritten as: Where Ψ is called stream function. The velocity field may be obtained from stream function as: Note that the stream function has a property that the flow normal to streamlines is zero. Hence, the fixed surfaces correspond to streamlines. Thus, the cylindrical surface IL may be treated as a streamline. Also, note that the velocity normal to the horizontal axis of symmetry is zero. Hence, the horizontal axis of symmetry may also be treated as a streamline. Similarly, the top surface (represented by line DH) is also a streamline. Figure TL07-2 Y ψ = 1 Since the velocity field depends on the relative difference of stream functions take the value of streamline along the horizontal axis of symmetry as zero, ψ = V Y o D H ψ = 0 i.e., ψ EI - ψ IL = 0 E ψ = 0 L I X Along the surface ED, u =V 0 = Velocity of flow Boundary Conditions ν = 0 COSMOSM Advanced Modules 6-17

56 Chapter 6 Verification Problems or ψ = -V 0 Y ψ DH - = -V 0 Y and ψ DH - = -1 Analogy between stream function formulation of potential flow and heat conduction. The governing equation of stream function formulation stream of potential flow is similar to steady state heat conduction equation with no heat generation. Head conduction Gradients of temperature Stream function Temperature Potential flow Velocity components Hence, HSTAR may be used to solve the potential flow problem by following the steps given below. 1. Set thermal conductivity Kx = Apply prescribed temperature boundary conditions wherever prescribed stream functions are to be applied. 3. The velocity field may be obtained by calculating the gradients of stream function (please see the options in PRINT command). COMPARISON OF RESULTS: At (x = 0, y = 0.1) (i.e., at Node 861). Stream Function Formulation Theory COSMOSM Ψ 0 0 u = ( Ψ/ y) υ = ( Ψ/ x) COSMOSM Advanced Modules

57 Part 1 HSTAR Heat Transfer Analysis TL08: Transient Heat Conduction in a Slab of Constant Thickness TYPE: Linear transient heat conduction, TRUSS2D elements. REFERENCE: Gupta, C. P., and Prakash, R., Engineering Heat Transfer, Nem Chand and Bros., India, 1979, pp Figure TL08-1 y PROBLEM: A large plate of thickness 62.8 cm is initially at a temperature of 50 C. Suddenly, both of its faces are raised to and held at 550 C. L Determine: 1. The Temperature at a plane 15.7 cm from the left surface, 5 hours after the sudden change in surface temperature. 2. Instantaneous heat flow rate at the left surface at the end of 5 hours. 3. Total heat flow across the surface at the end of 5 hours. Ts Ts X GIVEN: Thickness of slab = L = m Area of cross section = 1 m 2 Density = ρ = 23.2 Kg/m 3 Solution time = 5 hours Initial temperature = T i = 50 C Thermal conductivity = K = 46.4 J/m - hr K Specific heat = c = 1000 J/Kg - K Left and right surface temperatures = T s = 550 C Problem Sketch COSMOSM Advanced Modules 6-19

58 Chapter 6 Verification Problems MODELING HINTS: Since the other Figure TL08-2 dimensions of the plate are infinitely large, conduction occurs through thick ness, i.e., along x-axis. Therefore, this problem can be modeled with one dimensional elements having a total length of L Finite Element Model Temperature 1.0 (L = m) and 0.0 Time considering a cross 5.0 Temp._Time Curve sectional area of (A = 1 m 2 ). Sixteen TRUSS2D elements will be used to model this problem as shown in TL08-2. X ANALYTICAL SOLUTION Let: T = Temperature at any point x T s = Surface temperature T i = Initial temperature t = Time Temperature is: (n = 1, 3, 5, ----) Instantaneous heat flow rate per unit area at any point is: (n = 1, 3, 5, ----) 6-20 COSMOSM Advanced Modules

59 Part 1 HSTAR Heat Transfer Analysis Total heat flow during time t = 0 to t* is: (n = 1, 3, 5, ----) COMPARISON OF RESULTS: At time t* = 5 hours: Location Distance (m) Location Node No. Theory COSMOSM Difference % Temp (T) Heat Flow/ Unit Time (q) Cumulative Heat Flow (Q) , , ,125,330 1,092, Figure TL T e m p e r a t u r e Temperature Versus Time for Node Time COSMOSM Advanced Modules 6-21

60 Chapter 6 Verification Problems TL09: Heat Transfer from Cooling Fin TYPE: Heat transfer analysis, truss elements and convection link elements. REFERENCE: Kreith, F., Principles of Heat Transfer, International Textbook Co., Scranton, Pennsylvania, 2nd Printing, PROBLEM: A cooling fin of square crosssectional area A, length l, and conductivity k extends from a wall maintained at temperature T w. The surface convection coefficient between the fin and the surrounding air is h, the air temperature is T a, and the tip of the fin is insulated. Determine the heat conducted by the fin q and the temperature of the tip T l. Figure TL09-1 Z T w X h, T a b l b Z Y 15 Y GIVEN: b = 1 in = (1/12) ft Length of fin = l = ft Wall temperature = T w = 100 F Ambient temperature = T a = 0 F Film coefficient = h = 1 BTU/hr-ft 2 F Thermal conductivity = k = 25 BTU/hr-ft F Area of cross-section of the fin = ft COSMOSM Advanced Modules

61 Part 1 HSTAR Heat Transfer Analysis CALCULATED INPUT: The surface convection area per inch of fin length = ft 2. MODELING HINTS: The end convection elements are given half the surface area of the interior convection elements. Nodes 11 through 19 are given arbitrary locations. COMPARISON OF RESULTS: T at Node 9, F q at Node 1, Btu/hr Theory COSMOSM Difference 0.03% 0.13% COSMOSM Advanced Modules 6-23

62 Chapter 6 Verification Problems TL10: Temperature Distribution Due to Electrical Heating in a Wire TYPE: Steady state heat conduction with prescribed voltage and convection boundary conditions. REFERENCE: Rohsenow and Choi, Heat, Mass and Momentum Transfer. Figure TL10-1 D y C PROBLEM: Determine the temperature distribution in a current carrying wire. The voltage drop per foot of the wire is 0.1 volts. 1.0 ft GIVEN: Voltage on edge AB = 0 volts A B Voltage on edge DC = -0.1 volts ft Ambient temperature = 70 F Problem Sketch Thermal conductivity = 13 Btu/hr-ft F Electrical conductivity = E+7 mho/ft* Heat transfer coefficient on edge BC = 5 Btu/hr-ft 2 F x COMPARISON OF RESULTS: Temperature at node 1 = F (COSMOSM) = 420 F (Theory) The value of the electrical conductivity coefficient already contains the conversion factor from watt to Btu/hr COSMOSM Advanced Modules

63 Part 1 HSTAR Heat Transfer Analysis TL11: Temperature Distribution of Air Flowing Through a Pipe With a Constant Wall Temperature TYPE: Steady state fluid flow through a pipe using FLUIDT elements. The pipe is modeled with various types of elements as follows: TL11A 32 SHELL4 Elements TL11B 64 SHELL3 Elements TL11C 32 Solid Elements TL11D 189 TETRA10 Elements TL11E 189 TETRA4 Elements TL11F 400 TETRA4 Elements (finer mesh) REFERENCE: Rhosenow and Choi, Heat, Mass, and Momentum Transfer. PROBLEM: Find the temperature distribution of air flowing through a pipe whose wall is maintained at a constant temperature (same as problem TN06). Figure TL11-1 Mesh of a Quarter of a Pipe with Constant Wall Temperature and Air Flow (FLUIDT Elements) Node at inlet is outside the pipe (prescribed temperature) FLUIDT Elements COSMOSM Advanced Modules 6-25

64 Chapter 6 Verification Problems MODELING HINTS: Due to symmetry, only a quarter of the pipe circumference is modeled. Therefore, the pipe cross sectional area and the mass flow rate to be input for this analysis are 1/4 of the given values. Air flow in the pipe is modeled by 8 FLUIDT elements. The FLINKDEF command is used to associate the FLUIDT elements with the pipe wall for convection. The FLUIDT elements are generated by meshing a curve along the axis of the pipe. The curve is created such that its starting point falls outside the pipe (below the pipe inlet). As a result, the node associated with the starting point will not be considered for convection, and thus it can be assigned the inlet temperature. GIVEN: Temperature of the pipe wall = o C Temperature of air at Inlet = o C Pipe diameter = m Pipe length = m Mass flow rate = Kg/s Density of Air = Kg/m 3 Specific heat of air = J/ Kg o K Thermal conductivity of air = W/m o K Dynamic viscosity of air = 1.566E-5 Pa-s Parameters for evaluating Nusselt s number: C1 = 1.63 C2 = 0.08 C3 = 0.7 C4 = COSMOSM Advanced Modules

65 Part 1 HSTAR Heat Transfer Analysis COMPARISON OF RESULTS: Method/Element Type Temperature of Air at Pipe Outlet Error Percentage Theory 50.5 o C N/A COSMOSM (2D HLINK Elements (problem TN06) TL11A (3D FLUIDT Elements and 32 SHELL4 Elements) TL11B (3D FLUIDT Elements and 64 SHELL3 Elements) TL11C (3D FLUIDT Elements and 32 Solid Elements) TL11D (3D FLUIDT Elements and 189 TETRA10 Elements) TL11E (3D FLUIDT Elements and 189 TETRA4 Elements) TL11F (3D FLUIDT Elements and 400 TETRA4 Elements) o C % o C 1.3% o C 1.3% o C 1.3% o C 1.5% o C 0.9% o C 1.1% COSMOSM Advanced Modules 6-27

66 Chapter 6 Verification Problems TL12: Temperature Distribution in a Linear Accelerator with 3 Coolant Passages TYPE: Steady state linear heat conduction, heat convection, and fluid flow (SOLID + FLUIDT elements) REFERENCE: Los Alamos National Laboratory, LANSCE Division. PROBLEM: Find the temperature distribution in a Radio Frequency Quadrupole (RFQ) with an octagonal cross section and 3 coolant passages, due to heat flux applied to the surfaces of the accelerator cavity. The total applied heat flux is Btu per second for a length of 3 inches. Water, initially at room temperature, is used as the coolant flowing at a bulk velocity of 15 feet per second. The accelerator is made of copper. MODELING HINTS: The cross section of the model is shown in the figure. Due to symmetry, only 1/8 of the model is considered. Figure TL12-1. Cross Section of the Model Heat Flux Shaded area is 1/8 of the model Cooling Pipes 6-28 COSMOSM Advanced Modules

67 Part 1 HSTAR Heat Transfer Analysis To facilitate the application of the heat flux, a thin layer of SHELL4 elements was created (since some faces of the extruded solid elements may not be associated with the surfaces of the polyhedron). The FLINKDEF command is designed to ignore repeated areas for convection. Other data are taken from a recently completed accelerator and are therefore realistic. Figure TL12-2. Finite Element Model of 1/8 of a Linear Accelerator with 3 Coolant Passages Pipe A Pipe B Pipe C Figure TL12-2. Cross-section with Coolant Passages A, B, and C Figure TL12-3. Coolants Passages and Corresponding Convection Surfaces Heat Flux applied to surfaces COSMOSM Advanced Modules 6-29

68 Chapter 6 Verification Problems GIVEN: Applied Heat Flux Surface Heat Flux Density Heat Flux on Surface Surfaces 11 and Btu/s/in Btu/s Surface Btu/s/in Btu/s Surface Btu/s/in Btu/s Surface Btu/s/in Btu/s Surface Btu/s/in Btu/s Total Flux Btu/s Properties of Solid (Copper) Density = 0.84E-3 lbf.s 2 /in 4 Thermal Conductivity = 0.52E-2 Btu/in/s/ o F Specific Heat = 36.0 Btu.in/lbf/s 2 / o F Properties of Fluid (Water) Density = 0.93E-4 lbf.s 2 /in 4 (or lb/in 3 ) Thermal Conductivity = 0.82E-5 Btu/in/s/ o F Specific Heat = Btu.in/lbf/s 2 /F (or BTU/lb/ o F) Viscosity = 0.13E-6 lbf.s/in 2 (or lb/s/in) Temperature at inlet = 0.0 o F (with respect to reference) Section and Flow Properties for Passages A & B Diameter = 0.40 in Flow Area = in 2 Bulk Velocity = in/s Mass Flow rate = lbf.s/in (or lb/s = (Velocity)(Density)(Area) Film Coefficient =.0056 Btu/in 2 /s/ o F Section and Flow Properties for Passage C Diameter = 0.50 in Flow Area = in 2 Bulk Velocity = in/s Mass Flow rate = lbf.s/in (or lb/s) 6-30 COSMOSM Advanced Modules

69 Part 1 HSTAR Heat Transfer Analysis Film Coefficient =.0056 Btu/in 2 /s/ o F COMPARISON OF RESULTS: An energy balance between the applied energy (0.982 Btu/s) and the total energy gained by the coolant (water) gives a check on results: Theory COSMOSM Error = 0.916% de = (M a. dt a + M b. dt b + M c. dt c ) (g) = Btu/s T a = o F T b = o F T c = o F de = [(.51323) ( ) + (.57880) ( ) + (.41628) ( )] (386.) = Btu/s Where: T a is the temperature at the outlet of pipe A, T b is the temperature at the outlet of pipe B, and T c is the temperature at the outlet of pipe C. M a, M b, and M c are the mass flow rates for pipes A, B, and C. Figure TL12-4. Temperature Plot Pipe A Pipe B Pipe C COSMOSM Advanced Modules 6-31