Volume 11  Year 2024  Pages 5263
DOI: 10.11159/jffhmt.2024.006
HeatKernel Parametric Model of Heat Transfer Through Layered Materials Coupled to Heat Sinks
Edward Michaelchuck Jr.^{1}, Jesse Duncan ^{1}, Scott Ramsey ^{1}, Troy Mayo^{1}, Samuel Lambrakos^{1}
^{1}U.S. Naval Research Laboratory
4555 Overlook Ave SW, Washington, D.C. 20375
Edward.Michaelchuck@nrl.navy.mil; Jesse.Duncan@nrl.navy.mil;
Scott.Ramsey@nrl.navy.mil; Troy.Mayo@nrl.navy.mil;
Samuel.Lambrakos@nrl.navy.mil
Abstract  A parametric model of heat transfer through layered materials is presented. This model is formally based on the heatkernel solution to the advectiondiffusion equation, which is extended to include effects of multiple layers with varying thermal diffusivities, interface effects (e.g., large changes in thermal properties), contact resistance, and the effects of singular heatsinks, represented by negative heat sources. This model provides parametric representations of temperature distributions within layeredmaterial systems, which can be utilized for their design and optimization for layerconfiguration, including heatsink control of thermal transport. Results of prototype modelling of controlled heat transfer in layeredmaterial systems are presented and validated, demonstrating general aspects of the parametric model for thermal analysis and simulation of heattransfer control using layer configurations and embedded heat sinks.
Keywords: Heat Transfer model, multilayer materials, heat sinks, heat source distributions, basis function expansion.
© Copyright 2024 Authors  This is an Open Access article published under the Creative Commons Attribution License terms Creative Commons Attribution License terms. Unrestricted use, distribution, and reproduction in any medium are permitted, provided the original work is properly cited.
Date Received: 20230519
Date Revised: 20231214
Date Accepted: 20240313
Date Published: 20240326
1. Introduction
The control of heat transfer through multilayer materials is significant for many applications, ranging from heat treatment of materials to thermal management of systems. Optimizing heat transfer through multilayer materials requires estimating the thermal response of layered composite materials. This can be achieved using parametric models that combine heattransfer characteristics of a specified layered system and thermal material properties, enabling prediction of temperature fields. These models should be conveniently adaptable for estimating the thermal response of different types of layered materials. The present study presents a parametric model of heat transfer through a layered material system. The parametric model is motivated by welding processes [1], where work piece temperatures are controlled by material composition and thermal contact to base plates. The control of heat transfer through multilayer materials can also utilize heat sinks. The general physical character of heat sinks is that their thermal diffusivities are substantially greater than those of the work pieces whose temperature fields are to be controlled [2]. Accordingly, thermal coupling of heat sinks to adjacent layers can be represented parametrically by phenomenological negative heat sources [3].
Parametric modeling of temperature fields follows the approach of inverse analysis [4], i.e., inverse thermal analysis [510], which is for estimating optimal parameter values for a given system specification. The parameter values (e.g., effective thermal diffusivities) should be achievable for realistic system design. For complex material systems, the determination of material response functions is well posed in terms of inverse analysis. The multiscale character of layered materials poses an inverse analysis problem, which follows from the realization that layeredmaterial thermal properties are not the same as those of bulk materials. In addition, for layered materials there is typically influence on thermal transport due to advection at bounding surfaces of the layered system. Accordingly, parametrization should be extended to include effective diffusivities, which are based formally on replacing the advectiondiffusion operator with an effectivediffusion operator. Physically, advection is not expected to manifest as influencing thermal transport locally within a layeredmaterial system, but rather as influencing thermal transport over the its entire length of the layered system. Accordingly, the phenomenological influence of advection, which is associated with ambient environments at surface boundaries of a layeredmaterial system, again poses a problem of inverse thermal analysis for determination of effective diffusivities. Finally, heat transfer across interfaces of layers making up a layered material system can be effectively singular with respect to heattransfer trend characteristics because of thermal contact resistance and possible large differences of thermal diffusivity. Accordingly, with respect to parametric modeling, an interface may be represented by a layer having singular characteristics with respect to heat transfer. Heat transfer through a characteristically singular layer from adjacent layers of material will depend on the characteristic thermal coupling of layer interfaces, which again is a complex material property, not known a priori, and thus appropriately posed for inverse thermal analysis.
There exists different types of configurations for heatsink coupling to a heated system. The parametric model considered here, for heatsink coupling to a heated layered system, is described schematically in Figure 1, where coupling occurs at edges of the system. This configuration poses a specific problem with respect to inverse thermal analysis.
This study presents a parametric model for heat transfer through layered material systems, formally based on the heat kernel solution of the heat conduction equation [3], which includes effects of multiple layers with varying thermal diffusivities, interface effects (e.g., large changes in thermal properties), contact resistance, and the effects of singular heatsinks (represented by negative heat sources). This model is structured for analysis of both steady state and timedependent heat transfer in thin layered materials. In addition, this model is combined with another parametric model of heatsink cooling, which is based on a specific approximation of the heatkernel solution. Organization of subject areas presented are as follows. First, a parametric model of temperature fields for heat transfer through layer layered materials coupled to heat sinks is presented. Second, a formal mathematical analysis of the parametric model’s structure is presented, which provides physical interpretation of the parameters. Third, prototype simulations using the parametric model are described. Fourth, prototype inverse thermal analyses using the parametric model are presented. Finally, a discussion and conclusion are given..
2. Parametric Model
Presented in this section is a parametric model for heat transfer through a thin layered material heated at one surface, with transfer of heat to an ambient environment at the other surface by means of radiation and convection, with the possibility of heatsink channel cooling, at steady state. For brevity of the document and to assist reader comprehension, a nomenclature section, Section 9, outlines the various variables present in the parametric model and associated equations.
The parametric model is based formal extensions and approximations of analytical solutions to the heat conduction equation [3], is given by
and
The quantities h_{j}, j = 1N_{j}, are the interface contactdiffusivity parameters (analogous to contact conductance) whose units are 1/s, where locations of layerlayer interfaces are at x = x_{j}, and u(x) is the Heaviside unit step function. The source function f(x,y,z) is the 3D cooling field of the heat sink. The quantities τ_{H},C_{0}, C_{1}, Q_{x} Q_{y}, and Q which are formally a delay time, a basis function scaling parameter, another basis function scaling parameter, and cooling fluxes, which specify heat flux at the heating boundary (parameter ) and strength of coupling of heat sink layer to a cooling bath (parameter ) to adjacent material layers (parameters ). Note that physically, for a given location (x,y,z), heat sink cooling is formally for all x, y and z. Our goal, however, is parametric modelling of heat transfer to the surface of the layered system, and therefore the region x > x_{hs} is of most interest. The function f(x,y,z) assumes a model representation that is, in principle, based on either physical assumptions, i.e., materials having known thermal properties or phenomenological parameters adjusted with respect to measurements. Specifically, the parametric model is defined in terms of adjustable parameters, τ_{H}, C_{0}, C_{1}, Q_{x}, Q_{y}, Q_{z} and ∆T_{c} which are determined in principle according to experimental measurements, i.e., inverse thermal analysis. Similarly, contact conductances, h_{j,} are adjustable parameters to be determined by measurements.
Referring to Eqs. (4) and (5), note that 0 < yy_{hs} < L_{y} and 0 < zz_{hs} < L_{z} The quantity L_{z}, which for present simulations has arbitrary units, is scaled according to the transverse length l_{1} coupled to cooling bath as shown schematically in Figure 1. The quantity L_{y} (not considered here), together with L_{z}, represent twodimensional coupling to a cooling bath.
Next, given a surface boundary at x = x_{s}, and letting T_{S} = T(x_{s},y,z), defined by Eq. 1, for x > x_{s},
The quantities h_{c}, e, s, and T_{A} are the convective heattransfer coefficient, emissivity of the outer surface, StefanBoltzmann Constant (5.6704⸱10^{8} W⸱m^{2}⸱K^{4}) and the ambient temperature at the outer surface, respectively. Equations 16 define a parametric model for heat transfer through a layered material, from a hot body to a ambient environment external to the outer surface, with cooling due to an embedded heat sink.
3. Structure of the Parametric Model
This section examines the formal structure of the parametric model defined by Eqs. 16, which is for heat transfer through a thin layered material, heated at one surface, with transfer of heat to an ambient environment at the other surface by means of radiation and convection, with the possibility of heatsink channel cooling. Heat transfer is assumed either at steady state or time dependent. Time dependency of heat transfer is assumed to be measured and represented parametrically by means of timedelay parameters according to inverse thermal analysis. This model is based on formal extensions of heatkernel solution to the heat conduction equation [3], given by
Before proceeding with application of the model, there are key aspects of the type of layered system considered (see Figure 1), and its parametricmodel representation Eqs. 16, that should be emphasized for modelparameter interpretation. These are:
 The heatkernel Eq. 8 is a convenient ansatz for parametric representation of time evolution, i.e., timedependent heat transfer.
 For both steady state and timedependent heat transfer, the timedelay and coolingflux parameters can be given physical interpretations.
 The parametric model assumes that contact conductance between layers is dominant with respect to influencing heat transfer through the layered material system, relative to the influence of changes in thermal properties across an interface.
 The parametric model Eqs. 16 is convenient for parameter adjustment with respect to boundary conditions on the system shown in Figure 1.
 The thermal diffusivities can be either measured quantities (i.e., material properties), or themselves, adjustable phenomenological parameters.
The timedelay parameter may be interpreted by considering heat transfer to first order. Specifically, that the heat flux Q_{H} given by
for conductance k, integrated over interval (x, x_{s}) for a surface at T_{H}, gives
Next, Eq. 8 is reformulated to satisfy two boundary conditions, which are temperatures at the heating and outer surface boundaries of the layered material, and is thus
Next, assuming (x_{s} – x_{o}) is small, consistent with the condition of a thin layered material, it follows that
Comparing Eqs. 10 and 12,
It follows from Eq. 13 that the delaytime parameters, for “thin” layered materials (systems of our consideration) are inversely proportional to heat fluxes (or cooling fluxes), and accordingly, are physically consistent parameterizations of these fluxes. Finally, proceeding with analysis of Eq. 1, it can be seen that the source function f(x,y,z) is based on the approximation defined by Eqs. 12 and 13.
An important property of Eq. 1 is that, although it is structured for convenient encoding of timedependence, based on the underlying heatkernel ansatz, its mathematical property is that of tending toward piecewise linearity, i.e., formally Eq. 13. This is a manifestation of the underlying system, which is a “thin” layered material. This same property, tending to linearity, applies to Eqs. 35, representing heatsink coupling. The ansatz for these equations is the approximation defined by Eq. 13, which is convenient inverse thermal analysis of heatsink coupling to thin layered materials.
Temperature fields associated with both timedependent and steady state heat transfer over an extended spatial region, as in welding processes, based on their spatiotemporal characteristics, may be modelled parametrically using linear combinations of weighted heatkernel solutions, i.e., sums of heatkernel puffs. This type of representation requires adjusting sets of weight coefficients according to evolving temperature fields extending over space as a function of time, and eventually achieving steady state. The property of Eq. 1, of tending to piecewise linearity, implies that a single heatkernel puff is sufficient for parametric representation.
4. Prototype Simulations
This section presents simulations using the parametric model Eq. 16 of layerconfiguration and heatsink controlled thermal transport within a layered material system. The design of these simulations, using both physically realistic thermal properties and phenomenological adjustable parameters, was not to demonstrate optimal layerconfiguration and heatsink thermal control, but rather general characteristics of the parametric model for modelling and simulation of such control, as well as demonstrating feasibly of such control using multilayer and heatsink materials. The simulations described in this section may also be interpreted as representing prototype inverse thermal analyses, which are a goal of parametric modelling, in addition to estimating thermal response of a system given parameter values.
The simulations presented in this section where programmed in MATLAB©. The prototype simulations selected an example material stack up with known material properties for the various layers. The simulations assume a specific performance with regards to heat transfer and heat sink performance. The simulations use a time independent, isothermal boundary condition for the hot boundary and a time dependent isothermal boundary condition for the cold boundary in the absence of a heat sink. The cold boundary condition is primarily for parameterization of the system; therefore, in the presence of a heat sink, the cold boundary condition is not enforced – only the hot boundary condition is enforced.
Our first prototype simulations are of heat transfer from a hot body at 100^{o}C through a layered system, using the parametric model defined by Eqs. 17, which are shown in Figures 2 and 3. These simulations assume contact conductance between layers, represented by equivalent contact diffusivities, which have been determined by inverse thermal analysis. The temperature fields shown in Figures 2 and 3 are modified in computational experiments that follow. Shown in Figure 2 is the temperature field of the layered system at steady state. Shown in Figure 3 is the timedependent temperature field for evolution of this system to steady state. Referring to Figure 4, the model system considered here, neglecting ydependence of heatsink cooling represented by Eq. 3, is two dimensional and associated with four boundary conditions, which are specified at the heated and outersurface boundaries, and at the boundaries z = 0 and L_{z}. Again, the quantity L_{z}, which for present simulations has arbitrary units, is scaled according to the transverse length l_{1} coupled to cooling bath as shown schematically in Figure 1. The timedelay parameter, τ_{H} , which is equivalent to the hotboundary heat flux, is scaled with respect to measurements.
Our second prototype simulations are of heat transfer from a hot body through a layered system, where there is coupling of a highly conductive layer to a cooling bath (shown in Figures 5, 6, 7 and 8), using the parametric model defined by Eqs. 16. Note that the cooling layer is not represented explicitly, but phenomenologically. These simulations assume contact conductance between layers, and that heatsink coupling occurs at one or two of the boundaries along z, as shown in Figures 7 and 8, respectively. As indicated above, the model system, which is 2D, is defined by four boundary conditions. Referring to Figure 5, heat sink boundary conditions at steady state are those of an ambient cooling bath at 42 oC. Referring to Figure 6, heat sink boundary conditions for timedependent evolution to steady state are those of an ambient cooling bath at 0 oC.
With regards to Figures 5 and 6, the prototype simulations show the results of the multilayer system in the presence of a heat sink where Figure 5 is a steady state snap shot and Figure 6 is a time evolution of the system. The systems show that the heat sink draws energy and reduces the temperature of the materials in close proximity of the heat sink. This effect propagates through the various mediums, but primarily reduces the temperatures of the mediums farthest from the isothermal boundary condition. The model is qualitatively in agreement with circuit modelling approaches of heat transfer where the energy travels along the path of least resistance, i.e., the energy primarily travels from the hot boundary condition to the heat sink.
Referring to Figures 7 and 8, the model system represented by Eqs. 16 is two dimensional and associated with four boundary conditions, which are specified at the heated and outersurface boundaries, and at the boundaries z = 0 and L_{z}. Again, the quantity L_{z}, which for present simulations has arbitrary units, is scaled according to the transverse length l1 coupled to cooling bath (see Figure 1) and the parameter τ_{H} , is equivalent to the hotboundary heat flux, is scaled with respect to measurements.
The prototype simulations, whose results are shown in Figures 5 and 6, demonstrate the general procedure for inverse thermal analysis using the parametric model Eqs. 16. In reality, a layered material system, heatsink coupled to a cooling bath, is characterized by contact conductance’s at interfaces between layers (a function of the process used for layering) and anisotropic cooling along and transverse to the embedded heatsink channel. This anisotropic cooling depends on the nature of heatsinklayer coupling to adjacent layers and cooling bath. Accordingly, design of a heatsinkcooled layered material requires quantification of heatsink coupling that is convenient with respect to inverse thermal analysis. Equations 16 defines a parametric model requiring a relatively small number of parameters to be adjusted for encoding thermal response characteristics of embedded heat sinks. In essence, the parametric model Eqs. 16 adopts an ideal layered system, having no contact conductance, as an initial ansatz to be further adjusted according to inverse thermal analysis of heatsink coupling, where delaytime, τ_{H}, cooling fluxes Q_{x}, Q_{y} and Q_{z}, heatsink coupling parameter ∆T_{c},, are adjustable parameters, and contact diffusivities are determined by inverse thermal analysis. The parametric model Eqs. 16 is structured for 3D representation, and thus for the model system considered here, which is 2D, the parameter Q_{y} = 0.
5. Inverse Thermal Analysis
This section presents an inverse thermal analysis using the parametric model Eq. 16. To perform the inverse analysis, a series of temperature measurements must be collected in the system without heat sink coupling to create a series of equations and unknowns for the contact conductance’s and scaling parameters at steady state, Eqs. 14 – 15. It is assumed that contact conductance and the scaling parameters are not a function of time or heat transfer; thus, the model is analyzed at steady state. Due to the full length of Eq. 14, each complete equation for the specified boundary condition will be similarly separated into four pieces.
And
With the parameterization of Eq. 1, the heat sink can be coupled to a cooling to bath parameterize the heat sink terms, Q_{x}, Q_{y}, and Q_{z}, where the heat sink temperature change, ∆T_{c}, is experimentally measured. The quantities Q_{x}, Q_{y}, and Q_{z} are determined through a three point calibration. This results in a system of three equations and three unknowns, Eqs. 1518. Due to the full length of Eq. 14, each complete equation for the specified boundary condition will be similarly separated into four pieces.
6. Prototype Inverse Thermal Analysis
This analysis demonstrates the thermal response properties between layers and heatsink coupling with an emphasis on analysing a system with varying contact conductance between the layers. Because of their nature and the complexity as a function of specific system designs, the parametrization is not readily available based on formal thermal properties previously determined; therefore, the parameters must be determined inversely. For the purposes of this inverse analysis, a single multilayer material comprised of Nyco and copper will be considered where the copper acts as an embedded heat sink between two layers of Nyco. The material properties are listed in Table 1 which have been experimentally determined.
Table 1. Material properties for the cotton, copper, and Nyco system.
t [mm] 
𝜌 [Kg⸱m^{3}] 
k [W∙m^{1}∙K^{1}] 
C_{p} [J⸱kg^{1}⸱K^{1}] 
[m^{2}∙s^{1}] 

Copper 
0.051 
8960 
391.2 
389.4 
1.112⸱10^{4} 
Nyco 
0.356 
5.07 
15.4 
1445 
2.102⸱10^{3} 
To simplify the mathematical model and experimental test setup, a small sample with a cross section of 3 cm by 3 cm was evaluated where the temperature variation along the yaxis and zaxis were considered negligible. Additionally, the system was evaluated by sandwiching the samples in between an isothermal hot plate and cold plate where the system is tested with and without heat sink coupling, Figure 9. Due to the test setup, the radiation component was considered negligible. The resulting temperature data is listed in Table 2.
The experimental setup, Figure 9, has additional error sources; however, those error sources are inconsequential with respect to the validation of the parametric model. First, the experimental setup uses weighted blocks to compress the material stacks such that the contact between layers is consistent and repeatable – the physical contact between layers directly influences the thermal contact conductance between the layers. The second source of error stems from the thermistors and thermistor measurement controller. It is assessed that the thermistor and thermistor measurement controller combination have <0.1°C of temperature accuracy and variation based on previous validation measurements not discussed in this paper. For the isothermal hotplate, the temperature is controlled by a PID controller to minimize variation. Additionally, the hotplate is allowed to reach temperature prior to inserting the material and starting the experimental test. The final notable sources of error are the ambient environmental temperature and water bath temperature which were continuously monitored. Overall, these various sources of error are small with respect to the global temperature measurements within the material. Additionally, the model is parametrized via measurements using this machine such that errors of the machine measurements would equally present themselves in the model, but the relative error between the model and measurement would be consistent.
Table 2. Thermal testing for the cotton, copper, and Nyco system where a is the Hot Plate Temperature [^{o}C], b is the Interface 1 Temperature [^{o}C], c is the Interface 2 Temperature [^{o}C], and d is the Cold Plate Temperature [^{o}C].

[s] 
a [^{o}C] 
b [^{o}C] 
c [^{o}C] 
d [^{o}C] 
No Heat Sink 
0 
100.0 
19.1 
18.7 
18.7 
No Heat Sink 
1000 
100.0 
83.0 
78.8 
67.4 
Heat Sink 
0 
100.0 
19.1 
18.4 
18.4 
Heat Sink 
1000 
100.0 
85.2 
74.8 
65.3 
The thermal measurement results can be used in conjunction with Equations 1318 to parametrize the system via inverse analysis, which yields the basis function coefficients (C_{0} and C_{1}) and heat flux (Q_{x}), Table 3. This analysis assumes a 1D mapping of the 3D system such that the y and z component variation, Q_{y} and Q_{z}, is considered negligible; thus, evaluating to one. This results in Eq. 16’s simplification into Eq. 20.
Table 3. Parameterization of the NycoCopperNyco system.
System Parameter 
Parameterization Values 
C_{0} 
0.2179 
C_{1} 
7.2982∙10^{5} 
h_{0} 
1252.9193 
h_{1} 
48533.7643 
Q_{x}[K∙m^{1}] 
0.1934 
∆T_{C}[K] 
4.400 
Using the parameterization, the thermal model, Eqs. 14, can be evaluated to show the time evolution of the temperature fields with and without heat sink coupling, Figures 1012. Note that position zero represents the hot boundary, and the position 0.8 mm represents the cold boundary.
7. Discussion
The parametric model defined by Eqs. 16 is a formal adaptation of the heatkernel solution of the heatconduction equation for simulation of heattransfer in layered materials, which are coupled to heat sinks at edge boundaries. As described schematically in Figure 1, the model is of a threedimensional system that approximates heatsink cooling of a heated layered material. The purpose of which is to handle the multiscale nature of thin layered composites with embedded heat sinks. The parametric model combines both energytransport through an ideal system of layered materials, not having contact conductance, representing the model ansatz, and phenomenological parameterization for the influence of contact conductance and embedded heat sinks. The parametric model, whose construction is according to physical theory, includes the phenomenological source function f(x,y,z) representing multiscale coupling of embedded heat sinks, where cooling baths are located at edges of the system. Formulation of the source function f(x,y,z) is based on the approximation defined by Eqs. 11 and 12, which is formally reinterpreted with respect to a cooling flux at a boundary. This function is structured for encoding of thermal response associated with anisotropic heat transfer at steady state. Determination of anisotropic heat diffusion characteristics poses a problem for inverse thermal analysis. In the spirit of inverse analysis, parameterization of the model is not unique, but adapted for convenience.
The parametric model Eqs. 16 poses the problem of generating a parameter space that is sufficiently dense, expansive and bounded, for extraction of estimated parameter values for a given target layered system. Given a parameter space, estimation of target parameter values may be achieved by observation of parameter trends as a function of quantities characterizing a class of layered materials, e.g., material combinations, thickness, contact conductance, heatsink properties. Estimation of target parameter values can also be achieved using algorithms for searching within parameter spaces. Selection of appropriate parametersearch algorithms poses a problem itself, which should depend on the form of the parameter space.
Constructing the prototype experiment inverse analysis above exposed the singular nature of the heatsink and contact conductances relative to the other layers. It is apparent from Figures 1012 that there the model has excellent agreement with experimentally measured data, but there is a significant contact conductance singularity between the layers despite pressing the system together. This is to be expected as the contact conductance parameter encapsulates not only the insulated effect of poor layer contact, but also the reflected and transmitted heat waves not expressly given by this model.
8. Conclusion
Determination of optimal process parameters for achieving a given target temperature field for heat transfer through a layered material using materiallayer configurations and heat sinks poses a specific problem. The results of this study demonstrate use and general features of a parametric model, which can provide estimation of heattransfer characteristics for layered materials coupled to heat sinks while handling both the singular nature of the contact conductance’s and heat sinks. The nature of this model is that a singular basis function is used to represent the layered system while inverse analysis is performed at steady state. The model can be further modified to include a full basis function representation of all the layers using a discrete number of basis functions greater than one. This inherently will provide better estimation and agreement with experimentally measured results, but the increased complexity will complicate inverse thermal analysis. Additionally, as mentioned in the discussion, the contact conductances of this model inherently include the reflected and transmitted heat waves within the system. Thus, the model can be further modified to explicitly call out those parameters such that it further isolates only the effects of poor material contact between layers. Overall, the model, as presented in this analysis, provides sufficient agreement between experimentally measured data and simulated data for the analysis of thin, multiscale materials. The next steps will be to include more complex effects within the model and expand the number of basis functions to represent the time evolution of the fields.
9. Nomenclature
For brevity of the document, this nomenclature section outlines the various variables present in the parametric model and associated equations.
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