1. Thermal Resistance Network Analyzer

This analyzer solves steady state temperature distribution problems in thermal resistance networks. The analyzer resides in the mach_eval.analyzers.general.thermal_stator module.

1.1. Model Background

Thermal resistance networks are used to reduce the temperature distribution in a system into a set of nodes and thermal resistances. This is analogous to electrical resistance systems, where instead of V=IR it is dT=RQ where dT is the temperature rise, R is the thermal resistance, and Q is the heat flow.

This analyzer has the user specify fixed reference temperatures at one or more nodes, thermal resistances between nodes, and heat input at multiple nodes. The analyzer constructs the thermal network, solves it, and returns to the user the temperature at every node in the system. This analyzer is also utilized by other analyzers in eMach, i.e. SPM Rotor Thermal Analyzer.

1.1.1. Example Model

The image below shows an example thermal problem with its corresponding thermal resistance network overlayed on top.

The example consists of the following:

Three hardware components: a “base”, a thermal “conductor,” and a thermal “insulator”
Two sources of heat: Q₁at node 1 and Q₅at node 5
Convection on the right side of the model with a convection coefficient of h, represented by thermal resistances R_3,0and R_4,0
Conduction within and between the hardware components, represented by all other thermal resistances R.
Depth into the page of d, width w, and length L.

The code necessary to solve this example is developed in the sections below.

1.1.2. Resistances

The thermal_stator module contains a set of classes for creating conduction and convection thermal resistances for the analyzer. These classes are now specified.

These resistance classes all implement the Resistance protocol defined within the module. Classes are provided for geometry that is typical of electric machines, including cylinders and rectangles. The conduction resistances are implemented using standard heat flow expressions. Three specialized convection resistance classes air_gap_conv, hub_conv, and shaft_conv implement models presented in this paper:

1. 1. Howey, P. R. N. Childs and A. S. Holmes, “Air-Gap Convection in Rotating Electrical Machines,” in IEEE Transactions on Industrial Electronics, vol. 59, no. 3, pp. 1367-1375, March 2012.

All resistance classes initializers require at least three input arguments:

Node_1 and Node_2. These are int objects that indicate the nodes the resistance is connected between.
Mat. This is a Material object that holds the required material parameters. To initialize the Material class:
- Simple Conductor: my_mat = Material(k), where k is the material thermal conductivity in units of W/m-K.
- Fluid Material: my_mat = Material(k, cp, mu), where cp is the specific heat in units of kJ/kg and mu is viscosity in units of m^2/s. These parameters can be passed in as named arguments.

The following resistance classes are provided in the thermal_network module:

1.1.2.1. plane_wall

Class initializer signature: plane_wall(Material,Node_1,Node_2,L,A).

Node 1 is located on one of the walls perpendicular to the heat flow
Node 2 is located on the opposite face
L thickness of plane wall [m]
A cross-sectional area of plane wall [m²]

1.1.2.2. cylind_wall

Class initializer signature: cylind_wall(Material,Node_1,Node_2,R_1,R_2,H).

Node 1 is located at the inner surface of the cylinder
Node 2 is located at the outer cylinder.
R_1 radial location of node 1 [m]
R_2 radial location of node 2 [m]
H height of cylindrical wall [m]

1.1.2.3. air_gap_conv

Class initializer signature: air_gap_conv(Material,Node_1,Node_2,omega,R_r,R_s,u_z,A).

Node 1 is located on the surface of the inner cylinder
Node 2 is located in the air-gap fluid
omega rotational speed [rad/s]
R_r outer radius of rotor [m]
R_s inner radius of stator [m]
u_z axial airflow velocity [m/s]
A surface area of rotor [m²]

1.1.2.4. hub_conv

Class initializer signature: hub_conv(Material,Node_1,Node_2,omega,A).

Node 1 is located on the top surface of the cylinder
Node 2 is located in the fluid above the cylinder surface
omega rotational speed [rad/s]
A axial surface area of rotor [m²]

1.1.2.5. shaft_conv

Class initializer signature: shaft_conv(Material,Node_1,Node_2,omega,R,A,u_z).

Node 1 is located on the surface of the cylinder
Node-2 is located in the fluid.
omega rotational speed [rad/s]
R outer radius of shaft [m]
A radial surface area of rotor [m²]
u_z axial airflow velocity [m/s]

1.1.2.6. conv

Class initializer signature: conv(Material,Node_1,Node_2,h,A).

Node 1 is located on the surface
Node-2 is located in the fluid
h convection coefficient [W/m²-K]
A area of convection surface [m²]

1.2. Input from User

The analyzer problem initializer requires the user to provide the following information:

Resistances: List of Resistance objects.
Q_dot: List of heat sources for each node in unts of [W]. This list should be N_nodes in length, where the index of each entry determines which node the heat source is connected to.
T_ref: List of pairs of reference nodes and temperatures [[ref_node_1,ref_temp_1],[ref_node_2,ref_temp_2]..] in units of [C]
N_nodes: Number of nodes in the system

1.2.1. Example Code

Code is now provided to solve the example thermal problem provided in the Model Background section.

Import modules and define geometry.

import numpy as np
import scipy.optimize as op
from matplotlib import pyplot as plt
from matplotlib.patches import Rectangle
from eMach.mach_eval.analyzers.mechanical.thermal_network import *
#################
#Define Materials
#################

k_1=10 #Base Material Thermal Conductivity W/m-K
k_2=100 #Conductive Material Thermal Conductivity W/m-K
k_3=.01 #Insulating Material Thermal Conductivity W/m-K

mat1=Material(k_1)
mat2=Material(k_2)
mat3=Material(k_3)
##################
#Define Convection
##################
h=10 #Convection coefficient W/m^2-K
#################
#Define Geometry
#################

w=0.1 #Width m
L=.75 #Length m
d=.1 #Depth m

L_1=.5*L #Length of base
L_2=L*3/4 #Length to mid section 2 and 4
A_1=w*d #Cross sectional area of base
A_2=w*d/2 #Cross sectional area of section 2 and 3
A_3=(L-L_1)*d #Cross sectional Area between section 2 and 3

Create Resistance objects for this example.

N_nodes=6 #Number of Nodes

###################
#Define Resistances
###################
Resistances = []
##############
# Path 0
##############
Descr = "R_1,2"
Resistances.append(plane_wall(mat1, 1, 2, L_1, A_1))
Resistances[0].Descr = Descr
##############
# Path 1
##############
Descr = "R_2,3"
Resistances.append(plane_wall(mat2, 2, 3, L_2-L_1, A_2))
Resistances[1].Descr = Descr

##############
# Path 2
##############
Descr = "R_2,4"
Resistances.append(plane_wall(mat3, 2, 4, L_2-L_1, A_2))
Resistances[2].Descr = Descr

##############
# Path 3
##############
Descr = "R_3,5"
Resistances.append(plane_wall(mat2, 3, 5, w/4, A_3))
Resistances[3].Descr = Descr

##############
# Path 4
##############
Descr = "R_4,5"
Resistances.append(plane_wall(mat3, 4, 5, w/4, A_3))
Resistances[4].Descr = Descr

##############
# Path 5
##############
Descr = "R_3,0"
Resistances.append(conv(None, 3, 0, h, A_2))
Resistances[5].Descr = Descr

##############
# Path 6
##############
Descr = "R_4,0"
Resistances.append(conv(None, 4, 0, h, A_2))
Resistances[6].Descr = Descr

Specify the heat sources at nodes 1 and 5.

####################
#Define Heat Sources
####################
Q_dot=[0,]*N_nodes #create a list of 0's of length N_nodes
Q_dot[1]=10 #W
Q_dot[5]=10 #W

Specify the temperature at the reference node. For this example, only one reference temperatures is used (at node 0).

######################
#Define Reference Temps
######################
ref_node=0
ref_temp=25 #C
T_ref=[[ref_node,ref_temp],]

Create the problem and analyzer.

############################
#Create Problem and Analzyer
############################
prob=ThermalNetworkProblem(Resistances,Q_dot,T_ref,N_nodes)
ana=ThermalNetworkAnalyzer()

1.3. Output to User

The analyzer returns a list consisting of the temperature at each node of the resistance network in units of [C].

Example code to analyze the problem and graphically depict the temperature distribution of the nodes:

############################
#Analyze the Problem
############################
T=ana.analyze(prob)

############################
#Make Plot
############################
x=[L*1.2,0,L_1,L_2,L_2,L_2]
y=[0,0,0,w/4,-w/4,0]
fig,ax=plt.subplots(1,1)
c1=ax.scatter(x,y,c=T,s=200)
h=fig.colorbar(c1,label='Temperature')
# Create a Rectangle patch
rect = Rectangle((0,-w/2),L,w,linewidth=1,edgecolor='k',facecolor='none')
# Add the patch to the Axes
ax.add_patch(rect)
# Create a Rectangle patch
rect = Rectangle((L_1,0),L-L_1,w/2,linewidth=1,edgecolor='k',facecolor='none')
# Add the patch to the Axes
ax.add_patch(rect)
# Create a Rectangle patch
rect = Rectangle((L_1,-w/2),L-L_1,w/2,linewidth=1,edgecolor='k',facecolor='none')
# Add the patch to the Axes
ax.add_patch(rect)
ax.plot([x[1],x[2]],[y[1],y[2]],'r--')
ax.plot([x[2],x[3]],[y[2],y[3]],'r--')
ax.plot([x[2],x[3]],[y[2],y[4]],'r--')
ax.plot([x[3],x[5]],[y[3],y[5]],'r--')
ax.plot([x[4],x[5]],[y[4],y[5]],'r--')
ax.plot([x[3],x[0]],[y[3],y[0]],'r--')
ax.plot([x[4],x[0]],[y[4],y[0]],'r--')
ax.set_yticks([])
ax.set_xticks([])