The constant of proportionality, k, is the thermal conductivity BUT note that heat flows down a temperature gradient so we also introduce a negative sign to account for this and obtain: Thermodynamics
Thermal Conductivity k depends on the material and is called the thermal conductivity. Rearranging in terms of k we can evaluate the units of k: So k is defined as the rate of flow of heat through unit area of cross section of 1m of material when the temperature difference between the surfaces is 1K. Thermodynamics
Methods to measure thermal conductivity. Main methods: Steady state methods. Transient state methods
Steady state methods. Definition: These methods are used when the materials are in equilibrium state( means when temperature of the materials is constant). Advantage : Accurate readings can be taken. It steady state implies constant signals. Disadvantage: As material take to long time to reach equilibrium state so it is slow method.
TRANSIENT STATE METHOD: Definition: These methods are used during the heating of material. Advantage: Non-steady-state methods to measure the thermal conductivity do not require the signal to obtain a constant value. Readings can be taken during heating of material. Disadvantage: Readings are not accurate Mathematical analysis of the data is in general more difficult.
METHODS: IEEE Standard 442-1981, "IEEE guide for soil thermal resistivity measurements", ISBN 0-7381-0794-8. See also soil thermal properties.  IEEE Standard 98-2002, "Standard for the Preparation of Test Procedures for the Thermal Evaluation of Solid Electrical Insulating Materials", ISBN 0-7381-3277-2 ASTM Standard D5334-08, "Standard Test Method for Determination of Thermal Conductivity of Soil and Soft Rock by Thermal Needle Probe Procedure"  ASTM Standard D5470-06, "Standard Test Method for Thermal Transmission Properties of Thermally Conductive Electrical Insulation Materials"  ASTM Standard E1225-04, "Standard Test Method for Thermal Conductivity of Solids by Means of the Guarded-Comparative-Longitudinal Heat Flow Technique"  ASTM Standard D5930-01, "Standard Test Method for Thermal Conductivity of Plastics by Means of a Transient Line-Source Technique"  ASTM Standard D2717-95, "Standard Test Method for Thermal Conductivity of Liquids"  ISO 22007-2:2008 "Plastics -- Determination of thermal conductivity and thermal diffusivity -- Part 2: Transient plane heat source (hot disc) method" 
PROCEDUER: Take ice block ,find its weight and volume and place it on the material sheet. Then place it on the plate which is under observation Then provide steam for sometime then we the ice converted to water then measure the weight of the water (in beaker). Then place this values in the formula and calculate the conductivity.
Exp: Formula: K= Ro dx / A dT dx = thickness of material. dT =change in temperature.
RESEARCHES The record-setting anisotropic thermal conductivity of carbon nanotubes is enabling applications where heat needs to move from one place to another. carbon nanotubes have the intrinsic characteristics desired in material used as electrodes in batteries and capacitors.