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1 Introduction

Effective use of solar energy becomes important issues all over the world, since solar energy has been very promising as one of renewables which enable to reduce greenhouse gas emissions. Concentrating solar power (CSP) [1] is a system in which steam generated by solar heat drives a turbine to generate electricity. CSP is divided into two parts: one is the solar collector part and the other is power generation part. Direct normal irradiation in sunlight is only concentrated by the solar collector. The concentrated sunlight is converted into heat on the surface of receiver. For stable operation of CSP, the structure and materials in the solar collector part have to possess heat resistance and mechanical strength. Miyazaki Prefectural Government in Japan built a beam-down solar thermal collector at University of Miyazaki in 2012, the government is supporting and promoting the study and development of new energy. The collector [2] is the plant composed of reflectors and a receiver. Sunlight concentrated by reflectors which is referred to as heliostat is converted into heat on the surface of a receiver. Solar energy research project team in University of Miyazaki is working the study of receivers which enable them to convert concentrated sunlight into heat with high efficiency.

Before the beam-down solar thermal collector was established at University of Miyazaki, we considered a prototype of flat type receiver which is consisted of carbon steel fin and copper tubes. We focused on the effect of copper tubes on thermal conductance and mechanical strength at elevated temperatures in the receiver. To evaluate the prototype of the receiver, heating experiment was carried out by using a solar simulator [3]. A xenon short arc lamp of rating lamp input 500 W is used as a light source of the solar simulator. A xenon lamp is considered as a point light source with continuous spectra, which radiates light that is the closest of all artificial light sources to sunlight. An elliptical mirror is used to concentrate the light radiated by the lamp. When the light source in the lamp is set at the first focal point of the mirror, the radiated light which is reflected by the mirror is concentrated at the second focal point of the mirror. When the receiver is set at the second focal point of the mirror, the concentrated light absorbed by the receiver is converted into heat. The heat causes temperature rise in the receiver. Heat conducts along a copper tube in the receiver, and temperature rise of carbon steel fin on the copper tube can be observed. To elucidate heat conduction of the receiver from a theoretical viewpoint, we considered a compound flat plate composed of two different material layers as an analytical model, and one-dimensional heat conduction analysis was conducted as a first approximation [4]. When the numerical result of time-evolution of temperature on the top surface calculated by the theoretical analysis is compared with the experimental result, there is an inconsistency in time up to steady temperature state between the theoretical result and experimental one. Heat flows in the thickness direction in the one-dimensional heat conduction of the compound flat plate in which the top surface is subjected to uniform heat supply. Heat conduction from heated area to non-heated area cannot be considered in the one-dimensional heat conduction model of the plate.

The present study intends to evaluate heat conductance and mechanical strength at elevated temperature of a flat type receiver from a theoretical viewpoint. Considering a compound circular plate composed of two different kinds of material layers as an analytical model of the receiver, a mathematical analysis of plane axisymmetric transient heat conduction and thermal stresses for the plate is developed. It is assumed that the plate is subjected to heat flux on the top surface axisymmetrically and that the plate transfers heat with surrounding media. Analytical solutions of transient temperature change and thermal stresses are derived under a plane axisymmetric condition. Performing numerical calculations for the compound circular plate composed of carbon steel layer and copper one, the effect of thickness of carbon steel and copper layers on spatial variations and time-evolution of temperature change and thermal stresses in the plate is discussed briefly.

2 Analysis

2.1 Heat Conduction Problem

Consider a compound circular plate of radius a and of thickness B that is composed of two different kinds of material layers as an analytical model shown in Fig. 1.

Fig. 1
figure 1

Analytical model of compound circular plate

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The origin O of radial coordinate r and transverse coordinate z is set on the top surface of the plate. The layers are numbered, and thickness of i-th layer is b i ; local coordinate in the thickness direction whose origin is set on the top surface of the layer is z i ; temperature of the layer is θ i . Initial temperature in the plate is assumed to be constant temperature θ 0. Consider heat flux q(r, t) which has plane axisymmetric distribution is supplied on the top surface of the plate at time t. There are heat transfer between on the top, the bottom, and the side surfaces and surrounding media with temperatures θ a, θ b, θ s. Coefficients of heat transfer on the top, the bottom, and the side surfaces are γ a, γ b, γ s, respectively.

First, temperature change T i of each layer is defined as

$$\displaystyle{ T_{i}(r,z_{i},t) =\theta _{i}(r,z_{i},t) -\theta _{0}\;,\quad i = 1,2. }$$
(1)

The fundamental equation of transient heat conduction, the initial condition, the boundary conditions, and the continuity conditions in temperature change and heat flux are expressed in dimensionless forms as below:

$$\displaystyle\begin{array}{rcl} & & \frac{\partial \bar{T}_{i}} {\partial \tau } = \bar{\kappa }_{i}\left \{\frac{1} {\bar{r}} \frac{\partial } {\partial \bar{r}}\left (\bar{r}\frac{\partial \bar{T}_{i}} {\partial \bar{r}} \right ) + \frac{\partial ^{2}\bar{T}_{i}} {\partial \bar{z}_{i}^{2}} \right \}\;,\quad i = 1,2,{}\end{array}$$
(2)
$$\displaystyle\begin{array}{rcl} & & \bar{T}_{i} = 0\quad \mbox{ at}\quad \tau = 0\;,\quad i = 1,2,{}\end{array}$$
(3)
$$\displaystyle\begin{array}{rcl} & & -\frac{\partial \bar{T}_{1}} {\partial \bar{z}_{1}} + \bar{h}_{\text{a}}(\bar{T}_{1} -\bar{T}_{\text{a}}) = \frac{1} {\bar{\lambda }_{1}} f(\bar{r})g(\tau )\quad \mbox{ at}\quad \bar{z}_{1} = 0,{}\end{array}$$
(4)
$$\displaystyle\begin{array}{rcl} & & \frac{\partial \bar{T}_{2}} {\partial \bar{z}_{2}} + \bar{h}_{\text{b}}(\bar{T}_{2} -\bar{T}_{\text{b}}) = 0\quad \mbox{ at}\quad \bar{z}_{2} = \bar{b}_{2},{}\end{array}$$
(5)
$$\displaystyle\begin{array}{rcl} & & \frac{\partial \bar{T}_{i}} {\partial \bar{r}} + \bar{h}_{\text{s}i}(\bar{T}_{i} -\bar{T}_{\text{s}}) = 0\quad \mbox{ at}\quad \bar{r} = \bar{a}\;,\quad i = 1,2,{}\end{array}$$
(6)
$$\displaystyle\begin{array}{rcl} & & \bar{T}_{1} = \bar{T}_{2}\quad \mbox{ at}\quad \bar{z}_{1} = \bar{b}_{1}\,,\;\bar{z}_{2} = 0,{}\end{array}$$
(7)
$$\displaystyle\begin{array}{rcl} & & \bar{\lambda }_{1}\frac{\partial \bar{T}_{1}} {\partial \bar{z}_{1}} = \bar{\lambda }_{2}\frac{\partial \bar{T}_{2}} {\partial \bar{z}_{2}} \quad \mbox{ at}\quad \bar{z}_{1} = \bar{b}_{1}\,,\;\bar{z}_{2} = 0.{}\end{array}$$
(8)

Assuming the step heating shown in Fig. 2, functions of heat flux \(f(\bar{r})\) and g(τ) are expressed in dimensionless forms as

$$\displaystyle\begin{array}{rcl} f(\bar{r})& =& H(\bar{c} -\bar{ r}),{}\end{array}$$
(9)
$$\displaystyle\begin{array}{rcl} g(\tau )& =& H(\tau -\tau _{0}) - H(\tau -\tau _{0}-\varDelta \tau ),{}\end{array}$$
(10)

in which H( ) is Heaviside step function; \(\bar{c}\) is a dimensionless form of radius of heated area on the top surface of the plate c; τ, τ 0, Δ τ are dimensionless forms of time t, heating start time t 0, and the duration of heating Δ t.

Fig. 2
figure 2

Step heating

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Dimensionless quantities introduced in Eqs. (2)–(10) are defined by

$$\displaystyle{ \left.\begin{array}{rl} (\bar{z}_{i},\bar{b}_{i},\bar{r},\bar{a},\bar{c}) = \frac{(z_{i},b_{i},r,a,c)} {B},\;\bar{\kappa }_{i} = \frac{\kappa _{i}} {\kappa _{0}},\;\bar{\lambda }_{i} = \frac{\lambda _{i}} {\lambda _{0}} \\ \bar{q} = \frac{q} {q_{0}},\;T_{0} = \frac{q_{0}B} {\lambda _{0}},\;(\bar{T}_{i},\bar{T}_{\text{a}},\bar{T}_{\text{b}},\bar{T}_{\text{s}}) = \frac{(T_{i},T_{\text{a}},T_{\text{b}},T_{\text{s}})} {T_{0}} \\ (\tau,\tau _{0},\varDelta \tau ) = \frac{\kappa _{0}} {B^{2}} (t,t_{0},\varDelta t),\;(\bar{h}_{\text{a}},\bar{h}_{\text{b}},\bar{h}_{\text{s}i}) = B \times (h_{\text{a}},h_{\text{b}},h_{\text{s}i})\\ \end{array} \right \}, }$$
(11)

in which κ i and λ i are the thermal diffusivity and the heat conductivity of i-th layer; κ 0 and λ 0 are the typical values of the thermal diffusivity and the heat conductivity, respectively; q 0 is the intensity of heat flux; T 0 is the typical temperature change; h a, h b, and h si are relative heat transfer coefficients defined as

$$\displaystyle{ h_{\text{a}} = \frac{\gamma _{\text{a}}} {\lambda _{1}} \,,\;h_{\text{b}} = \frac{\gamma _{\text{b}}} {\lambda _{1}} \,,\;h_{\text{s}i} = \frac{\gamma _{\text{s}}} {\lambda _{i}} \;,\quad i = 1,2. }$$
(12)

Analytical solution of temperature change in the plate is obtained by the superposition of both solution to the heat conduction problem such that the plate is subjected to heat flux at top surface whose intensity varies with time and solution to the heat conduction problem such that the plate has heat transfer between the boundary surfaces and surrounding media.

$$\displaystyle\begin{array}{rcl} \bar{T}_{i}& =& \sum \limits _{k=1}^{\infty }\,T_{ ik}J_{0}(q_{ik}\bar{r})\;,\quad i = 1,2,{}\end{array}$$
(13)
$$\displaystyle\begin{array}{rcl} T_{ik}& =& \frac{2q_{ik}^{2}} {\bar{a}^{2}(q_{ik}^{2} + h_{\text{s}i}^{2})J_{0}^{2}(q_{ik}\bar{a})}\left \{-\sum \limits _{j=1}^{m}\frac{2\mu _{j}I_{\text{T}}(\tau )} {\varDelta '(\mu _{j})} [\bar{A}_{\text{u}\,ikj}\cosh (\alpha _{ikj}\bar{z}_{i}) + \bar{B}_{\text{u}\,ikj}\sinh (\alpha _{ikj}\bar{z}_{i})]\right. \\ & & -\sum \limits _{j=m+1}^{\infty }\frac{2\mu _{j}I_{\text{T}}(\tau )} {\varDelta '(\mu _{j})} [\bar{A}_{\text{u}\,ikj}\cos (\beta _{ikj}\bar{z}_{i}) + \bar{B}_{\text{u}\,ikj}\sin (\beta _{ikj}\bar{z}_{i})] \\ & & +\sum \limits _{j=1}^{m}\frac{2e^{-\mu _{j}^{2}\tau }} {\mu _{j}\varDelta '(\mu _{j})} [\bar{A}_{ikj}\cosh (\alpha _{ikj}\bar{z}_{i}) + \bar{B}_{ikj}\sinh (\alpha _{ikj}\bar{z}_{i})] \\ & & +\sum \limits _{j=m+1}^{\infty }\frac{2e^{-\mu _{j}^{2}\tau }} {\mu _{j}\varDelta '(\mu _{j})} [\bar{A}_{ikj}\cos (\beta _{ikj}\bar{z}_{i}) + \bar{B}_{ikj}\sin (\beta _{ikj}\bar{z}_{i})] \\ & & \left.+ \frac{1} {D}[\bar{A}'_{ik}\cosh (q_{ik}\bar{z}_{i}) + \bar{B}'_{ik}\sinh (q_{ik}\bar{z}_{i}) + D\,\frac{\bar{a}\bar{h}_{\text{s}i}\bar{T}_{\text{s}}} {q_{ikj}^{2}} J_{0}(q_{ik}\bar{a})]\right \}. {}\end{array}$$
(14)

In Eqs. (13) and (14), J 0( ) is the Bessel function of the first kind of order zero; I T (τ) is defined as

$$\displaystyle{ I_{\text{T}}(\tau ) =\int _{ 0}^{\tau }e^{-\mu _{j}^{2}(\tau -\tau ') }[H(\tau ' -\tau _{0}) - H(\tau ' -\tau _{0}-\varDelta \tau )]\;\text{d}\tau '. }$$
(15)

The details of symbols in Eq. (14) are omitted due to lack of space.

2.2 Thermal Stress Problem

We now consider stress and bending deformation in the compound circular plate subjected to thermal loads. If the origin of the coordinate in the thickness direction is appropriately selected in the cross-section of the plate, thermal bending of the plate can be treated easily. Thus, the coordinate in the thickness direction z′ whose position of the origin is located at z = η from the top surface z = 0 is defined as

$$\displaystyle{ \bar{z}' =\bar{ z}-\bar{\eta }, }$$
(16)

When the Poisson ratios of two layers assume to be same value ν, the position η of the origin of the coordinate z′ is defined as

$$\displaystyle{ \bar{\eta }= \bar{B}_{1}\left /\right. \bar{A}_{1}, }$$
(17)

in which \(\bar{\eta }\) is dimensionless form of the position of neutral plane of the plate η. It is assumed that Kirchhoff–Love hypothesis holds for the bending deformation of the plate subjected to plane axisymmetric heating. When body forces are absent, the equilibrium equations of the plate under plane axisymmetric condition are expressed in dimensionless forms in terms of resultant forces N r , N θ and resultant moments M r , M θ as

$$\displaystyle\begin{array}{rcl} \frac{\text{d}\bar{N}_{r}} {\text{d}\bar{r}} + \frac{\bar{N}_{r} -\bar{N}_{\theta }} {\bar{r}} & =& 0,{}\end{array}$$
(18)
$$\displaystyle\begin{array}{rcl} \frac{\text{d}^{2}\bar{M}_{r}} {\text{d}\bar{r}^{2}} + \frac{2} {\bar{r}} \frac{\text{d}\bar{M}_{r}} {\text{d}\bar{r}} -\frac{1} {\bar{r}} \frac{\text{d}\bar{M}_{\theta }} {\text{d}\bar{r}} & =& 0.{}\end{array}$$
(19)

Dimensionless forms of resultant forces \(\bar{N}_{r}\), \(\bar{N}_{\theta }\) and resultant moments \(\bar{M}_{r}\), \(\bar{M}_{\theta }\) are expressed as

$$\displaystyle{ \left.\begin{array}{r} \bar{N}_{r} = \bar{A}_{1}\left (\frac{\text{d}\bar{u}_{r0}} {\text{d}\bar{r}} +\nu \frac{\bar{u}_{r0}} {\bar{r}} \right ) -\bar{N}_{\text{T}} \\ \bar{N}_{\theta } = \bar{A}_{1}\left (\nu \frac{\text{d}\bar{u}_{r0}} {\text{d}\bar{r}} + \frac{\bar{u}_{r0}} {\bar{r}} \right ) -\bar{N}_{\text{T}} \\ \bar{M}_{r} = -\left (\bar{D}_{1} -\frac{\bar{B}_{1}^{2}} {\bar{A}_{1}} \right )\left (\frac{\text{d}^{2}\bar{w}} {\text{d}\bar{r}^{2}} +\nu \frac{1} {\bar{r}} \frac{\text{d}\bar{w}} {\text{d}\bar{r}} \right ) -\bar{M}_{\text{T}} \\ \bar{M}_{\theta } = -\left (\bar{D}_{1} -\frac{\bar{B}_{1}^{2}} {\bar{A}_{1}} \right )\left (\nu \frac{\text{d}^{2}\bar{w}} {\text{d}\bar{r}^{2}} + \frac{1} {\bar{r}} \frac{\text{d}\bar{w}} {\text{d}\bar{r}} \right ) -\bar{M}_{\text{T}}\\ \end{array} \right \}, }$$
(20)

in which \(\bar{u}_{r0}\) and \(\bar{w}\) are dimensionless forms of radial displacement on the neutral plane u r0 and deflection w; \(\bar{A}_{1}\), \(\bar{B}_{1}\), and \(\bar{D}_{1}\) are dimensionless forms of rigidities of the plate A 1, B 1, and D 1; \(\bar{N}_{T}\) and \(\bar{M}_{T}\) are dimensionless forms of thermal resultant force N T and thermal resultant moment M T . Their details are omitted due to lack of space.

Substituting Eq. (20) into the equations of equilibrium (18) and (19), and solving them under the restraint condition in deflection at the center of the plate and the mechanical boundary conditions

$$\displaystyle{ \left.\begin{array}{rl} \bar{w} = 0\quad \text{at}\quad \bar{r} = 0\\ \bar{N} _{r } = 0, \bar{M} _{r } = 0\quad \text{at} \quad \bar{r} =\bar{ a}\\ \end{array} \right \}, }$$
(21)

radial displacement on the neutral plane and deflection are expressed in dimensionless forms as

$$\displaystyle\begin{array}{rcl} \bar{u}_{r0}& =& \frac{1} {\bar{A}_{1}}\sum \limits _{k=1}^{\infty }\sum \limits _{ j=1}^{2}\frac{F_{jk}} {q_{ik}} \left \{ \frac{1} {1-\nu }\right.J_{1}(q_{ik}\bar{r})\left.+ \frac{1} {1+\nu } \frac{\bar{r}} {\bar{a}}J_{1}(q_{ik}\bar{a})\right \},{}\end{array}$$
(22)
$$\displaystyle\begin{array}{rcl} \bar{w}& =& \frac{\bar{A}_{1}} {\bar{A}_{1}\bar{D}_{1} -\bar{ B}_{1}^{2}}\sum \limits _{k=1}^{\infty }\sum \limits _{ j=1}^{2} \frac{G_{jk}} {q_{ik}^{2}}\left [ \frac{1} {1-\nu }\right.\left \{J_{0}(q_{ik}\bar{r}) - 1\right \}\left.- \frac{1} {1+\nu } \frac{J_{0}(q_{ik}\bar{a})} {\bar{a}} \frac{\bar{r}^{2}} {2} \right ]. \\ & & {}\end{array}$$
(23)

Then, dimensionless forms of thermal stress components \(\bar{\sigma }_{rri}\), \(\bar{\sigma }_{\theta \theta i}\) of i-th layer are expressed as

$$\displaystyle\begin{array}{rcl} \bar{\sigma }_{rr\,i}& =& \frac{\bar{E}_{i}} {1 -\nu ^{2}}\sum \limits _{k=1}^{\infty }\sum \limits _{ j=1}^{2}\left (\frac{F_{jk}} {\bar{A}_{1}} + \frac{\bar{A}_{1}G_{jk}\bar{z}'} {\bar{A}_{1}\bar{D}_{1} -\bar{ B}_{1}^{2}}\right )\left \{\frac{J_{0}(q_{ik}\bar{r})} {1-\nu } \right. \\ & & \left.- \frac{1} {q_{ik}}\left (\frac{J_{1}(q_{ik}\bar{r})} {\bar{r}} -\frac{J_{1}(q_{ik}\bar{a})} {\bar{a}} \right )\right \} -\frac{\bar{\alpha }_{i}\bar{E}_{i}} {1-\nu }\bar{T}_{i}, {}\end{array}$$
(24)
$$\displaystyle\begin{array}{rcl} \bar{\sigma }_{\theta \theta \,i}& =& \frac{\bar{E}_{i}} {1 -\nu ^{2}}\sum \limits _{k=1}^{\infty }\sum \limits _{ j=1}^{2}\left (\frac{F_{jk}} {\bar{A}_{1}} + \frac{\bar{A}_{1}G_{jk}\bar{z}'} {\bar{A}_{1}\bar{D}_{1} -\bar{ B}_{1}^{2}}\right )\left \{\frac{\nu J_{0}(q_{ik}\bar{r})} {1-\nu } \right. \\ & & \left.+ \frac{1} {q_{ik}}\left (\frac{J_{1}(q_{ik}\bar{r})} {\bar{r}} -\frac{J_{1}(q_{ik}\bar{a})} {\bar{a}} \right )\right \} -\frac{\bar{\alpha }_{i}\bar{E}_{i}} {1-\nu }\bar{T}_{i}, {}\end{array}$$
(25)

in which \(\bar{\sigma }_{rri}\) and \(\bar{\sigma }_{\theta \theta i}\) are dimensionless forms of radial and circumferential components of thermal stress σ rri , σ θ θ i of i-th layer; \(\bar{E}_{i}\) and \(\bar{\alpha }_{i}\) are dimensionless forms of the Young’s modulus E i and the coefficient of linear thermal expansion α i of i-th layer. Details of coefficients F jk and G jk are omitted.

Dimensionless quantities introduced in Eqs. (16)–(25) are defined as

$$\displaystyle{ \left.\begin{array}{r} (\bar{z},\bar{z}',\bar{\eta }) = \frac{(z,z',\eta )} {B},\;(\bar{u}_{r0},\bar{w}) = \frac{(u_{r0},w)} {\alpha _{0}T_{0}B},\;\bar{E}_{i} = \frac{E_{i}} {E_{0}},\;\bar{\alpha }_{i} = \frac{\alpha _{i}} {\alpha _{0}} \\ (\bar{N}_{r},\bar{N}_{\theta },\bar{N}_{\text{T}}) = \frac{(N_{r},N_{\theta },N_{\text{T}})} {\alpha _{0}E_{0}T_{0}B},\;(\bar{M}_{r},\bar{M}_{\theta },\bar{M}_{\text{T}}) = \frac{(M_{r},M_{\theta },M_{\text{T}})} {\alpha _{0}E_{0}T_{0}B^{2}} \\ (\bar{\sigma }_{rr\,i},\bar{\sigma }_{\theta \theta \,i}) = \frac{(\sigma _{rr\,i},\sigma _{\theta \theta \,i})} {\alpha _{0}E_{0}T_{0}},\;\bar{A}_{1} = \frac{A_{1}} {E_{0}B},\;\bar{B}_{1} = \frac{B_{1}} {E_{0}B^{2}},\;\bar{D}_{1} = \frac{D_{1}} {E_{0}B^{3}}\\ \end{array} \right \}, }$$
(26)

in which E 0 and α 0 are typical values of the Young modulus and the coefficient of linear thermal expansion.

3 Numerical Results and Discussion

Assuming heating experiment of the compound circular plate composed of carbon steel and copper layers using a solar simulator, numerical calculations are performed by making use of analytical solutions formulated in Sect. 2. The effect of thickness of copper layer on the spatial variations and time-evolutions of the temperature change and the thermal stress is examined.

Parameters in numerical calculation are given by

$$\displaystyle{ \left.\begin{array}{l} a = 50\,\text{mm},\;c = 15\,\text{mm,}\;\varDelta t = 600\,\text{s},\;q_{0} = 92.4\,\text{kW}\left /\right. \text{m}^{2} \\ b_{1} = 0.3\,\text{mm},\;\theta _{0} = 25\,^{\circ }\text{C,}\;\theta _{\text{a}} = \theta _{\text{b}} = \theta _{\text{s}} = 30\,^{\circ }\text{C} \\ \gamma _{\text{a}} = \gamma _{\text{b}} = \gamma _{\text{s}} = 50\,\text{W}\left /\right. \text{m}^{2}\text{K}\\ \end{array} \right \}, }$$
(27)

in which b 1 is thickness of carbon steel layer; c is radius of region which is subjected to heat supply; Δ t is the duration of heating; and q 0 is the intensity of heat flux, respectively. The thickness of copper layer b 2 was changed parametrically as b 2 = (0. 0 , 0. 3 , 1. 2 , 2. 1 , 3. 0) mm. The material properties of carbon steel and copper are shown in Table 1.

Table 1 Material properties
Full size table

Figure 3 shows the effect of thickness of copper layer on radial variations of temperature on the top surface z = 0 mm at time t = 600 s. As the thickness of copper layer increases, temperature rise over the area of the plate which is subjected to heat supply decreases, meanwhile temperature rise around the area increases. Temperature has a maximum at the center of the plate r = 0 mm, and has a minimum at the circumference r = 50 mm. As the thickness of copper layer increases, the maximum temperature decreases, and the difference between maximum temperature and minimum one becomes small. The heat conductivity of copper layer is larger than that of carbon steel, which raises the temperature around the area which is not subjected to heat supply.

Fig. 3
figure 3

Effect of thickness of copper layer on radial variation of temperature on the top surface z = 0 mm at time t = 600 s

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Figure 4 shows the effect of thickness of copper layer on time-evolution of temperature at the center on the top surface r = 0 mm, z = 0 mm. As the thickness of copper layer increases, time up to steady temperature state becomes longer and maximum temperature decreases.

Fig. 4
figure 4

Effect of thickness of copper layer on time-evolution of temperature at the center on the top surface r = 0 mm, z = 0 mm

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Figure 5 shows the effect of thickness of copper layer on the radius of the area in which temperature is above 110 ∘C. As the thickness of copper layer increases, the radius of the area increases. However, the radius of the area decreases, when the thickness exceeds a certain value. That means that there is an optimal thickness of the copper layer so as to maximize the radius of the area in which temperature is above a certain value. These results indicate that we may determine the thickness of the copper layer in the compound plate so as to maximize the temperature and the radius of the area.

Fig. 5
figure 5

Effect of thickness of copper layer on the radius of the area in which temperature is above 110 ∘C

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Figure 6 shows the effect of the thickness of copper layer on the radial variation of the circumferential component of thermal stress σ θ θ on the top surface z = 0 mm at time t = 600 s. When the thickness of copper is zero millimeters, compressive stress occurs in the heated area and tensile stress occurs in the unheated area. As the thickness of copper layer increases, the radial variation of the stress becomes flat and the maximum compressive stress at the center on the top surface.

Fig. 6
figure 6

Effect of thickness of copper layer on the radial variation of thermal stress σ θ θ on the top surface z = 0 mm at time t = 600 s

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Figure 7 shows the effect of thickness copper layer on the maximum compressive thermal stress σ θ θ on the top surface of the plate z = 0 mm. Maximum compressive stress occurs at the center on the top surface r = 0 mm at time t = 600 s when temperature has maximum value. As the thickness of copper layer increases, the maximum compressive thermal stress σ θ θ decreases monotonously.

Fig. 7
figure 7

Effect of thickness of copper layer on the maximum compressive thermal stress σ θ θ on the top surface z = 0 mm at time t = 600 s

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4 Conclusion

In the present study, a mathematical analysis of plane axisymmetric transient thermal stress of a circular plate composed of two different material layers is developed. Analytical solutions of temperature change, radial displacement at a neutral plane, deflection, and thermal stresses are derived under mechanical boundary condition of free surface at the whole side edge. Performing numerical calculation for a compound circular plate consisted of carbon steel layer and copper one, the effect of thickness of copper layer is discussed.

Numerical results on the effect of thickness of copper layer on spatial variations and time-evolutions of temperature change and thermal stresses are summarized as follows:

  1. 1.

    As the thickness of copper layer increases, time up to steady temperature state becomes longer and maximum temperature at the center of the plate decreases.

  2. 2.

    As the thickness of copper layer increases, temperature rise over the area of the plate which is subjected to heat supply decreases, meanwhile temperature rise around the area increases.

  3. 3.

    There is an optimal thickness of the copper layer so as to maximize the radius of the area in which temperature is above a certain value.

  4. 4.

    As the thickness of copper layer increases, the radial variation of the stress becomes flat and the maximum compressive stress at the center on the top surface of the plate.

  5. 5.

    As the thickness of copper layer increases, maximum temperature change and maximum compressive stress at the center on the top surface of the plate decreases, and the radial variations become flat.

The information obtained from this study can give an instruction for design of receiver which converts concentrated sunlight into heat with high efficiency and possesses mechanical strength at elevated temperature.