The governing thermal equations were derived from rst principles to create the continu- ous thermal pipe model. A continuous model was rst derived before the complex task of semi-discretising the model as described in Section 3.6.
The continuous thermal pipe model was derived from rst principles and used to describe the temperatures at points through the lateral or radial section of a pipe wall. The in- ternal copper points were assumed to adhere to a two-dimensional conduction equation. Where the copper material interfaces with other materials (e.g. water or air) then other thermal equations are required to describe the thermal ux which occurs at these points. boundary conditions are required to equate the diering thermal behaviours between dif- ferent materials at an interface. Boundary conditions are required for any isolated system to account for energy transfer through the section to be simulated and the interaction with the conditions outside the isolated system.
CHAPTER 3. THERMAL MODELLING 30 ˙ qV = 2rq˙A ˙ qA= −kc∂T∂r ˙ qA ˙q = hair(T − T∞) Copper Water Air r = Do/2 r = Di/2 T∞ T (z, r, t) Tw(z, t) r z vw(t)
Figure 3.1: Continuous thermal pipe model diagram. It can be seen that the model describes the copper which forms the pipe wall (shown in the magni ed section of the pipe wall to illustrate the energy transfers which occur). The inner and outer pipe diameters, Di and Do, show interfaces between materials. The heat ux, ˙q, which
occurs at each interface is shown as well as the temperature, T , of each material.
Figure 3.1 shows the notation used for the continuous thermal pipe model. It can be seen that the governing equations to describe the spatially continuous temperature can be represented separately for each material (water, copper and air).
3.5.1 Copper
The copper pipe was modelled as having a wall thickness of 1 mm and length of 1 m (minimum expected length of copper pipe in installations). The copper was assumed to only conduct heat for internal points for the normal case. This assumption is generally considered a valid because the copper is a solid.
The general heat conduction equation (which described the thermal behaviour of the copper for no external energy input) was derived
∂T
∂t = α∇
2T, α = kc
ρccpc (3.3)
where α, kc, ρc and cpc are the thermal diusivity, thermal conductivity, density and speci c heat capacity of copper respectively. Equation (3.3) can be expanded to cylindrical coordinates in two-dimensions to more speci cally describe the outlet pipe geometry to form Equation (3.4). ∂T ∂t = α ∂2T ∂z2 + α 1 r ∂ ∂r r∂T ∂r (3.4) Equation (3.4) can be simpli ed because the curvature of the pipe changes insigni cantly over the width of the pipe. This simpli cation was to facilitate computational e ciency in later stages. The computationally inexpensive form of Equation (3.4) is:
∂T ∂t = α ∂2T ∂z2 + α ∂2T ∂r2 (3.5)
Equation (3.5) forms the continuous equation which describes all internal (copper) points of the thermal pipe model. The section of pipe which is being modelled is not in nitely large, and pipe section interacts with external conditions which have dierent thermal properties. Where the outermost copper pipe points interface with external conditions (e.g. ambient air, or water interfaces) then energy exchanges with the external points occur which do not strictly follow the thermal behaviour dictated by Equation (3.5). Boundary conditions must be established at these interfacing surfaces to describe the energy exchanges which occur.
3.5.2 Boundary Conditions
Boundary conditions are necessary to account for the physical exchanges which occur between the interior points (which are described by Equation (3.5)) and the surrounding environment (which are not described by (3.5)). Boundary conditions enable a section of a system to be simulated without having to simulate the entire system.
3.5.2.1 Inner Pipe Boundary
The hot water interfaces with the internal copper wall on the inner pipe boundary. Several assumptions were made regarding the simulated behaviour of the water. This was done to avoid the complexities associated with the simulation of turbulent uids which is required to simulate the heat of a moving uid. The water is simulated as a one-dimensional convection equation with heat ux into the internal wall of the copper pipe wall. The water assumptions are:
Constant radial temperature (Tw(z, r, t) = Tw(z, t)) Incompressibility (vw(z, t) = vw(t), ρw and cpw constant)
Su cient and constant turbulence for considered velocities (fully developed turbu- lent ow)
The continuous form of the energy balance in the water medium is shown in Equation (3.6) where v is the velocity of water and ρw, cpware the density and speci c heat capacity of water respectively. The volumetric heat ux out of the water (into the internal pipe wall), ˙qV, must be equal to the radial conduction into the copper pipe wall to avoid energy accumulation at the interface.
ρwcpw ∂Tw
∂t = −vρwcpw ∂Tw
∂z − ˙qV (3.6)
Performing an energy balance equating the volumetric heat ux ˙qV (from Equation (3.6)) to the surface heat ux ˙qA allows for a conversion between the dierent measures of heat ux.
CHAPTER 3. THERMAL MODELLING 32
˙
qV dz πr2 = ˙qAdz 2πr ∴ ˙qV = 2
rqA˙ (3.7)
Equation (3.7) shows the relationship between volume and surface heat ux where dz is the longitudinal dierential element. Equation (3.6) can be rewritten using Equation (3.7) to represent the water convection equation with surface heat ux energy input:
ρwcpw∂Tw ∂t = −vρwcpw ∂Tw ∂z − 2 rqA˙ ∴ ∂Tw ∂t = −v ∂Tw ∂z − 1 ρwcpw 2 rqA˙ (3.8)
The heat ux occurring at the water-copper interface must be equal to avoid energy accumulation at the interface. The surface heat ux between the water at the interface must be equal to the conduction heat ux between the interface and the internal copper wall. Thus the water surface heat ux, ˙qA, can be equated to copper thermal conduction as they must be equal.
˙
qA = −kc∂T
∂r (3.9)
Equation (3.9) shows the heat ux which occurs at the copper-water interface, where kc is the thermal conductivity of copper. Equation (3.8) can be rewritten using Equation (3.9) into a convenient form to implement boundary conditions.
∂Tw ∂t = −v ∂Tw ∂z − 1 ρwcpw 2 r(−kc ∂T ∂r) = −v∂Tw ∂z + 1 ρwcpw 2 rkc ∂T ∂r (3.10)
Equation (3.10) was used to equate the one-dimensional water convection to the internal copper points at the water-copper interface because the temperatures and ux at any interface must be equal.
3.5.2.2 Outer Pipe Boundary
The outer pipe boundary is generally exposed to air, but several conditions can occur which need to be described by speci c equations. The options for outer boundary con- ditions are commonly air convection, insulated, heat source or heat sink. The general convection equation is given by Equation (3.11) where ˙q is the heat ux, h is the heat transfer coe cient and T∞ the temperature of ambient air:
All outer boundary conditions must occur at the air interface with conductive copper surface, therefore each scenario must somehow be equivalent to the temperature and heat ux at the conductive copper interface. An approximation can be made to represent all outer boundary conditions using the convection equation by changing the heat transfer coe cient in the general convection Equation (3.11) to the relevant coe cient for dierent media for certain sections of pipe (i.e. hair, hheatsink). This approximation is not precise, but the simulation of convection of uids is complex enough to justify making such a broad assumption which allows the behaviour of turbulent uids to be avoided. This simpli cation to a single convective case allows the heat ux due to conduction in the copper at the interface to be equated to the heat ux due to convection in the air at the interface. It also enables for dierent lengths of the pipe to have dierent convective interfaces applied.
The convection equation in air is the most common condition for the system, and shown in Equation (3.12).
˙q = hair(T − T∞) (3.12)
The heat ux occurring in the copper at the copper-air interface is shown in Equation (3.13).
− kc∂T
∂r = ˙q (3.13)
To avoid energy accumulation at the interface both heat ux values must again be equal. The heat ux found in equations (3.12) and (3.13) must be equal.
˙q = −kc∂T
∂r = hair(T − T∞) (3.14)
Equation (3.14) accounts for the external copper wall-air interface. 3.5.2.3 Fixed Temperature
When points on an interface are assumed to be maintained at a xed temperature (e.g. if a control system is in place to achieve this using a heating element and sensor) then these points are no longer represented as variables in the system. The speci c points are instead replaced with a constant temperature value and treated accordingly. This xed temperature condition was useful to simulate the pipe boundary nearest the EWH when the EWH was assumed to be at a xed temperature (because always-on setpoint control was implemented).