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| 1 | +# Heat Transfer in Fluids - Theory Manual and User Guide |
| 2 | + |
| 3 | +This document describes the fluid heat solver of Range FEA - `RSolverFluidHeat`, |
| 4 | +which serves the **Heat transfer in fluids** problem type: the equation it |
| 5 | +solves, the meaning of every input it accepts, its coupling to the heat solver |
| 6 | +of a solid, and the parts of the graphical user interface that drive it. |
| 7 | + |
| 8 | +Heat transfer in fluids is solved on the flow field of an **Incompressible |
| 9 | +viscous flow** task. The flow solver, its boundary conditions, its non-linear |
| 10 | +iteration and its time march are described in the [Fluid flow](theory_fluid.md) |
| 11 | +manual, and the heat solver of solids in the [Heat transfer](theory_heat.md) |
| 12 | +manual. |
| 13 | + |
| 14 | +**Contents** |
| 15 | + |
| 16 | +1. [Theoretical background](#1-theoretical-background) |
| 17 | +2. [Graphical user interface](#2-graphical-user-interface) |
| 18 | +3. [Limitations](#3-limitations) |
| 19 | + |
| 20 | +--- |
| 21 | + |
| 22 | +## 1. Theoretical background |
| 23 | + |
| 24 | +### 1.1 Primary unknown - the temperature |
| 25 | + |
| 26 | +The primary unknown is the **temperature** `T` in `K`, one degree of freedom per |
| 27 | +node. |
| 28 | + |
| 29 | +| Problem type | Solver | Primary unknown | Units | DOF per node | |
| 30 | +|---|---|---|---|---| |
| 31 | +| Heat transfer in fluids | `RSolverFluidHeat` | temperature | `K` | 1 | |
| 32 | + |
| 33 | +The problem type **requires** an *Incompressible viscous flow* task in the same |
| 34 | +problem: it reads the velocity field from its result and never computes it |
| 35 | +itself. |
| 36 | + |
| 37 | +### 1.2 Governing equation |
| 38 | + |
| 39 | +**Heat transfer in fluids** solves the advection-diffusion of temperature on the |
| 40 | +flow field, |
| 41 | + |
| 42 | +``` |
| 43 | +rho * c * ( dT/dt + v . grad(T) ) = div( k * grad(T) ) + q |
| 44 | +``` |
| 45 | + |
| 46 | +with the same SUPG stabilisation of the convective term as the flow solver and |
| 47 | +the same theta time march. The velocity `v` is **recovered from the flow |
| 48 | +result**, not solved here - which is why the problem type requires an |
| 49 | +*Incompressible viscous flow* task. |
| 50 | + |
| 51 | +| Symbol | Meaning | Units | Source | |
| 52 | +|---|---|---|---| |
| 53 | +| `T` | temperature | `K` | solved for | |
| 54 | +| `v` | velocity | `m/s` | the flow result | |
| 55 | +| `k` | thermal conductivity | `W/(m*K)` | material property | |
| 56 | +| `c` | heat capacity | `J/(kg*K)` | material property | |
| 57 | +| `rho` | density | `kg/m^3` | material property | |
| 58 | +| `q` | heat source density | `W/m^3` | Heat condition, Joule heat, radiation | |
| 59 | + |
| 60 | +The difference from the plain **Heat transfer** problem type is exactly the |
| 61 | +`v . grad(T)` term: energy is carried along by the moving fluid rather than only |
| 62 | +conducted. That also means this solver has no convection boundary conditions - |
| 63 | +there is nothing to correlate, since the heat leaving a wall into the fluid is |
| 64 | +resolved by the mesh rather than modelled. |
| 65 | + |
| 66 | +### 1.3 Boundary conditions |
| 67 | + |
| 68 | +| Boundary condition | Type | Applies to | Components | |
| 69 | +|---|---|---|---| |
| 70 | +| Temperature | explicit (Dirichlet) | point, line, surface, volume | Temperature `[K]` | |
| 71 | +| Heat | natural, source | point, line, surface, volume | Heat `[W]` | |
| 72 | + |
| 73 | +**Temperature** holds the nodes of its entity at the given value. Assign it to |
| 74 | +the inlet surface to set the temperature of the fluid entering the domain, and |
| 75 | +to a wall to model a heated or cooled boundary. |
| 76 | + |
| 77 | +**Heat** is a source and takes the **total power in `W`** delivered into the |
| 78 | +entity, spread over the measure of the entity before assembly - the same |
| 79 | +convention as the heat solver. *Heat rate (unit area)* and *Heat rate (unit |
| 80 | +volume)* are offered by the interface for this problem type but are **not read** |
| 81 | +by the fluid heat solver; use *Heat* for a source here. |
| 82 | + |
| 83 | +Every component is a **table against time**. |
| 84 | + |
| 85 | +The flow conditions - walls, inlets and outlets - belong to the flow task and |
| 86 | +are described in the [Fluid flow](theory_fluid.md) manual. |
| 87 | + |
| 88 | +### 1.4 Conjugate heat transfer |
| 89 | + |
| 90 | +A wall between a meshed fluid and a meshed solid carrying the **Forced |
| 91 | +convection** condition of the plain *Heat transfer* solver couples the two. Once |
| 92 | +the heat solver has run, the fluid heat solver holds the wall nodes at the |
| 93 | +temperature of the solid - until then the wall is insulated - and hands back, |
| 94 | +for every wall element, the pair `(h, Tf)` that reproduces the heat flux its |
| 95 | +solution takes through the wall. The heat solver applies that pair as a |
| 96 | +convection condition. The flux is the residual of the fluid equations at the wall |
| 97 | +nodes, consistent with the discretisation, so it stays accurate in a thin thermal |
| 98 | +boundary layer. The wall temperature is relaxed with an Aitken factor from pass |
| 99 | +to pass, which settles the exchange in a handful of passes. The |
| 100 | +[Heat transfer](theory_heat.md) manual describes the scheme under *Conjugate |
| 101 | +heat transfer*. Radiative heat and Joule heat arriving from other solvers are |
| 102 | +added to the source term the same way as in the heat solver. |
| 103 | + |
| 104 | +### 1.5 Convergence |
| 105 | + |
| 106 | +Uncoupled, the fluid heat solver reports itself as converged unconditionally - |
| 107 | +it is linear once the flow field is fixed, so one solve is the final answer. A |
| 108 | +group holding both the flow and the fluid heat task therefore ends when the flow |
| 109 | +task converges. With coupled walls it reports the relative change of its |
| 110 | +temperature field instead, so the group iterates until the wall temperature has |
| 111 | +settled. |
| 112 | + |
| 113 | +### 1.6 Initial conditions |
| 114 | + |
| 115 | +| Initial condition | Effect | |
| 116 | +|---|---| |
| 117 | +| Temperature | initial temperature field | |
| 118 | + |
| 119 | +Initial conditions are read on the **first run only**; after that the field |
| 120 | +marches from the previous result. A boundary condition overrides an initial |
| 121 | +condition where the two meet. |
| 122 | + |
| 123 | +### 1.7 Results |
| 124 | + |
| 125 | +| Result | Apply to | Meaning | |
| 126 | +|---|---|---| |
| 127 | +| Temperature `[K]` | node | the solved temperature field | |
| 128 | +| Heat flux `[W/m^2]` | element | conductive flux `-k*grad(T)` | |
| 129 | + |
| 130 | +--- |
| 131 | + |
| 132 | +## 2. Graphical user interface |
| 133 | + |
| 134 | +Everything relevant lives in the **Solver** dock (`Problem`, |
| 135 | +`Boundary conditions`, `Initial conditions`, `Material`, `Results` tabs), the |
| 136 | +**Problem** menu and the **Model** dock. |
| 137 | + |
| 138 | +### 2.1 Selecting the problem |
| 139 | + |
| 140 | +`Problem` -> `Problem(s) task flow` (`Ctrl+P`) opens the problem task flow |
| 141 | +dialog. The combinations worth knowing: |
| 142 | + |
| 143 | +- *Incompressible viscous flow* and *Heat transfer in fluids* in one group - a |
| 144 | + heated or cooled flow; |
| 145 | +- *Heat transfer in fluids* and a separate *Heat transfer* task on a meshed |
| 146 | + solid, the fluid heat task first, in one group of several iterations - |
| 147 | + conjugate heat transfer through the walls carrying *Forced convection*. |
| 148 | + |
| 149 | +The problem type requires a flow task to be present. The task group values - |
| 150 | +**# of iterations** and **Convergence** - are described in the |
| 151 | +[Fluid flow](theory_fluid.md) manual. |
| 152 | + |
| 153 | +### 2.2 Problem tab |
| 154 | + |
| 155 | +**Time-solver** - shown since the problem type is time-dependent. Leave it |
| 156 | +disabled for a steady-state run. Its fields are described in the |
| 157 | +[Fluid flow](theory_fluid.md) manual. |
| 158 | + |
| 159 | +The problem type has no setup group box of its own. Everything else is assigned |
| 160 | +per entity in the condition tabs. |
| 161 | + |
| 162 | +### 2.3 Material tab |
| 163 | + |
| 164 | +| Property | Units | Comment | |
| 165 | +|---|---|---| |
| 166 | +| Density | `kg/m^3` | `rho`, about `1000` for water, `1.2` for air | |
| 167 | +| Dynamic viscosity | `kg/(m*s)` | `mu`, about `1.0e-3` for water, `1.8e-5` for air | |
| 168 | +| Thermal conductivity | `W/(m*K)` | `k`, about `0.6` for water, `0.025` for air | |
| 169 | +| Heat capacity | `J/(kg*K)` | `c`, about `4180` for water, `1005` for air | |
| 170 | + |
| 171 | +An entity missing any required property is not solved, and the setup checker |
| 172 | +warns before the run starts. The shipped material database includes **Water** |
| 173 | +and **Air**, which is usually the whole of this step. Every property is a table |
| 174 | +against temperature, evaluated at the element temperature of the previous run. |
| 175 | + |
| 176 | +### 2.4 Boundary conditions tab |
| 177 | + |
| 178 | +**Temperature** and **Heat** appear once a *Heat transfer in fluids* task is in |
| 179 | +the task flow - see section 1.3. Every component is a table against time; click |
| 180 | +**Edit time dependent values** to enter values against time. |
| 181 | + |
| 182 | +### 2.5 Initial conditions |
| 183 | + |
| 184 | +The `Initial conditions` tab offers *Temperature*. It sets the starting field of |
| 185 | +the first run. |
| 186 | + |
| 187 | +### 2.6 Matrix solver setup |
| 188 | + |
| 189 | +`Problem` -> `Setup Problem(s) matrix solver` configures the iterative solvers. |
| 190 | +The fluid heat solver uses the **GMRES** entry, since its matrix is not |
| 191 | +symmetric. |
| 192 | + |
| 193 | +### 2.7 Monitoring points and results |
| 194 | + |
| 195 | +`Problem` -> `Define monitoring points` can record *Temperature*. The `Results` |
| 196 | +tab shows temperature as a node scalar and heat flux as an element vector. |
| 197 | + |
| 198 | +--- |
| 199 | + |
| 200 | +## 3. Limitations |
| 201 | + |
| 202 | +- **No buoyancy.** The density does not depend on temperature, so a *Heat |
| 203 | + transfer in fluids* result never drives the flow it was computed on. Natural |
| 204 | + convection, thermal plumes and stratification cannot be modelled. |
| 205 | +- **The coupling to the flow is one-way.** The flow drives the heat transport; |
| 206 | + the temperature never changes the flow. |
| 207 | +- **`Heat rate` conditions are ignored.** *Heat rate (unit area)* and *Heat rate |
| 208 | + (unit volume)* are offered by the interface for *Heat transfer in fluids* but |
| 209 | + are not read by `RSolverFluidHeat`; only the *Heat* condition reaches it. The |
| 210 | + plain *Heat transfer* problem type does read all three. |
| 211 | +- **No convection boundary conditions.** Heat leaving a wall into a meshed fluid |
| 212 | + is resolved rather than correlated, so the *Simple*, *Forced* and *Natural |
| 213 | + convection* conditions belong to the plain heat solver. Conjugate heat |
| 214 | + transfer is set up as a *Heat transfer* task on the solid whose *Forced |
| 215 | + convection* walls read the computed fluid state. |
| 216 | +- **No turbulence model.** The flow is laminar, see the |
| 217 | + [Fluid flow](theory_fluid.md) manual, so turbulent heat transfer is not |
| 218 | + represented. |
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