Derivation of the heat transfer matrix inside ISO 13786

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The purpose of this article is to provide the derivation of the heat transfer matrix of ISO 13786 from the solution of the one-dimensional heat equation (partial differential equation, PDE). The matrix refers to building components as defined in ISO 13786. Inside the same regulation the heat transfer matrix is given without a proof and no derivation is available online.

In the one-dimensional heat equation considered in this case, T(x,t) is the temperature which is a function of the x-coordinate and time.

(1) $\begin{equation*}\frac{\partial T}{\partial t} = \frac{\lambda}{c\rho} \frac{ \partial^2 T}{\partial x^2} \end{equation*}$

This partial differential equation is solved by the so-called method of separation of variables. The solution is supposed to be made of the product of two other functions of one of the variables x and t separately:

(2) $\begin{equation*} T(x,t) = T_0 + U(t) \cdot X(x) \end{equation*}$

The constant inside the T(x, t) function is neglected:

(3) $\begin{equation*} \Delta T(x,t) = T(x,t) - T_0 = U(t) \cdot X(x) \end{equation*}$

The PDE becomes then:

(4) $\begin{equation*}\frac{\partial \Delta T}{\partial t} = \frac{\lambda}{c\rho} \frac{ \partial^2 \Delta T}{\partial x^2} \end{equation*}$

Now, the T(x,t) function is replaced by the two separate functions inside the PDE and the product rule is employed:

(5) $\begin{equation*} X(x) \cdot U'(t) = \frac{\lambda}{c\rho} U(t) \cdot X''(x) \end{equation*}$

The above equation is rearranged as follows:

(6) $\begin{equation*} \frac{X''(x)}{X(x)} = \frac{c\rho}{\lambda} \frac{U'(t)}{U(t)} = \mu \end{equation*}$

We assume now that μ is a complex number with the real part equal to 0.

(7) $\begin{equation*} \mu \in, \mathbb{C}, \Re(\mu) = 0 \end{equation*}$

Now the above PDE can be split into two separate ordinary differential equations (ODE) which can be solved separately:

(8) $\begin{equation*} \begin{cases} X''(x) = \mu X(x) \\ U'(t) = \frac{\lambda \mu}{c \rho} U(t)\end{cases}\end{equation*}$

(9) $\begin{equation*} \begin{cases} X(x) = A e^{\sqrt{\mu} x} + A e^{-\sqrt{\mu} x} \\ U(t) = B e^{\frac{\lambda \mu}{c \rho} t}\end{cases}\end{equation*}$

(10) $\begin{equation*} \mu = k \cdot j \end{equation*}$

(11) $\begin{equation*} \begin{cases} X(x) = A e^{\sqrt{k j} x} + B e^{-\sqrt{k j} x} \\ U(t) = C e^{\frac{\lambda k j}{c \rho} t} = B e^{j \omega t}\end{cases}\end{equation*}$

We introduce the new variable ω (angular velocity), which is equal to:

(12) $\begin{equation*} \omega = \frac{\lambda k}{c \rho} \implies k = \frac{\omega c \rho}{\lambda} \end{equation*}$

The power of the exponentials inside the X(x) function can be rewritten as:

(13) $\begin{equation*} \sqrt{k j} = \sqrt{\frac{\omega c \rho}{\lambda} j} = \sqrt{\frac{\omega c \rho}{\lambda} e^{j \frac{\pi}{2}}} = \sqrt{\frac{\omega c \rho}{\lambda}} e^{j \frac{\pi}{4}} = \sqrt{\frac{\omega c \rho}{\lambda}} \left ( cos \left (\frac{\pi}{4} \right ) + j sin \left ( \frac{\pi}{4} \right ) \right ) = \end{equation*}$

(14) $\begin{equation*} = \sqrt{\frac{\omega c \rho}{\lambda}} \left ( \frac{1}{\sqrt{2}} + j \frac{1}{\sqrt{2}} \right ) = \sqrt{\frac{\omega c \rho}{2\lambda}} ( 1 + j) = \frac{1}{\delta} (1+j) \end{equation*}$

δ is defined as:

(15) $\begin{equation*}\delta = \sqrt{\frac{2\lambda}{\omega c \rho}}\end{equation*}$

The two differential equations above can be written as:

(16) $\begin{equation*} \begin{cases} X(x) = A e^{\frac{1}{\delta} (1+j) x} + B e^{-\frac{1}{\delta} (1+j) x} \\ U(t) = C e^{j \omega t}\end{cases}\end{equation*}$

Now, the solution of the one-dimensional heat equation can be written:

(17) $\begin{equation*} \Delta T(x,t) =\left ( A e^{\frac{1}{\delta} (1+j) x} + B e^{-\frac{1}{\delta} (1+j) x} \right ) C e^{j \omega t} = \left ( A' e^{\frac{1}{\delta} (1+j) x} + B' e^{-\frac{1}{\delta} (1+j) x} \right ) e^{j \omega t} \end{equation*}$

where

(18) $\begin{equation*} A' = A \cdot C\end{equation*}\begin{equation*} B' = B \cdot C\end{equation*}$

The following parameters are introduced:

θ₀: maximum temperature at side n of the building component
q₀: maximum heat flux density at side n of the building component
θ_L: maximum temperature at side m of the building component
q_L: maximum heat flux density at side m of the building component

The origin of the x-axis is located on side n while the x-value on the opposite side m is equal to L. Therefore, the solution can be written on the two boundaries as:

(19) $\begin{equation*} \Delta T(0,t) = \theta_0 e^{j \omega t} = (A' + B') e^{j \omega t} \implies \end{equation*}$

(20) $\begin{equation*} \implies \theta_0 = (A' + B') \end{equation*}$

(21) $\begin{equation*} q(0,t) = q_0 e^{j \omega t} = -\lambda \frac{\partial T}{\partial x} \bigg|_{x=0} = -\lambda \left { \left [ A' \frac{1}{\delta} (1+j) e^{\frac{1}{\delta} (1+j) x} - B' \frac{1}{\delta} (1+j) e^{- \frac{1}{\delta} (1+j) x} \right ] e^{j \omega t} \right } \bigg|_{x=0} = \end{equation*}$

(22) $\begin{equation*} = - \frac{\lambda}{\delta} (1+j) (A'-B') e^{j \omega t} \implies q_0 = \frac{\lambda}{\delta} (1+j) (A'-B') \end{equation*}$

(23) $\begin{equation*} \Delta T(L,t) = \theta_L e^{j \omega t} = \left [ A' e^{\frac{1}{\delta} (1+j) L} + B' e^{-\frac{1}{\delta} (1+j) L} \right ] e^{j \omega t} \implies \end{equation*}$

(24) $\begin{equation*} \implies \theta_L = \left [ A' e^{\frac{1}{\delta} (1+j) L} + B' e^{-\frac{1}{\delta} (1+j) L} \right ] \end{equation*}$

(25) $\begin{equation*} q(L,t) = q_{L} e^{j \omega t} = -\lambda \frac{\partial T}{\partial x} \bigg|_{x=L} = -\lambda \left { \left [ A' \frac{1}{\delta} (1+j) e^{\frac{1}{\delta} (1+j) x} - B' \frac{1}{\delta} (1+j) e^{- \frac{1}{\delta} (1+j) x} \right ] e^{j \omega t} \right } \bigg|_{x=L} \end{equation*}$

(26) $\begin{equation*} \implies q_{L} = -\lambda \left [ A' \frac{1}{\delta} (1+j) e^{\frac{1}{\delta} (1+j) L} - B' \frac{1}{\delta} (1+j) e^{- \frac{1}{\delta} (1+j) L} \right ] \end{equation*}$

The four equations obtained are as follows:

(27) $\begin{equation*} \begin{cases} \theta_0 = (A' + B') \\ q_0 = \frac{\lambda}{\delta} (1+j) (A'-B') \\ \theta_L = \left [ A' e^{\frac{1}{\delta} (1+j) L} + B' e^{-\frac{1}{\delta} (1+j) L} \right ] \\ q_{L} = -\lambda \left [ A' \frac{1}{\delta} (1+j) e^{\frac{1}{\delta} (1+j) L} - B' \frac{1}{\delta} (1+j) e^{- \frac{1}{\delta} (1+j) L} \right ] \end{cases}\end{equation*}$

Now we rearrange the first two equations so that parameters A’ and B’ are isolated on left-hand side of each of the two equations:

(28) $\begin{equation*} \begin{cases} A' = \frac{\theta_0}{2} - \frac{q_0}{4\lambda} \delta (1-j) \\ B' = \frac{\theta_0}{2} + \frac{q_0}{4\lambda} \delta (1-j) \end{cases}\end{equation*}$

Those two parameters are then replaced in the third and fourth equations:

(29) $\begin{equation*} \begin{cases} \theta_L = cosh \left ( \frac{L}{\delta} (1+j) \right ) \theta_0 - \frac{\delta}{2 \lambda} (1-j) sinh \left ( \frac{L}{\delta} (1+j) \right ) q_0 \\ q_L = - \frac{\lambda}{\delta} (1+j) sinh \left ( \frac{L}{\delta} (1+j) \right ) \theta_0 + cosh \left ( \frac{L}{\delta} (1+j) \right ) q_0 \end{cases} \end{equation*}$

The following properties of hyperbolic functions with complex argument are used in order to obtain the final form of the matrix provided inside regulation ISO 13786:

(30) $\begin{equation*} sinh(z_1 + z_2) = sinh(z_1)cosh(z_2) + cosh(z_1)sinh(z_2) \end{equation*} \begin{equation*} cosh(z_1 + z_2) = cosh(z_1)cosh(z_2) + sinh(z_1)sinh(z_2) \end{equation*} \begin{equation*} cosh(ix) = cos(x) \end{equation*} \begin{equation*} sinh(ix) = i \cdot sin(x) \end{equation*}$

After a few calculations and after relacing parameter ξ, defined as:

(31) $\begin{equation*} \xi = \frac{L}{\delta} \end{equation*}$

the final form is obtained:

(32) $\begin{equation*} \left [ \begin{array}{c} \theta_L \\ q_L \end{array} = \left [ \begin{array}{cc} M_{1,1} & M_{1,2} \\ M_{2,1} & M_{2,2} \end{array} \right ] \cdot \left [ \begin{array}{c} \theta_0 \\ q_0 \end{array} \right ] \end{equation*}$

where:

(33) $\begin{equation*} M_{1,1} = M_{2,2} = cosh(\xi)cos(\xi) + j sinh(\xi) sin(\xi) \end{equation*} \begin{equation*} M_{1,2} = \frac{\delta}{2\lambda} \left [ sinh(\xi)cos(\xi) + cosh(\xi)sin(\xi) + j \left ( cosh(\xi) sin (\xi) - sinh(\xi) cos(\xi) \right ) \right ] \end{equation*} \begin{equation*} M_{2,1} = - \frac{\lambda}{\delta} \left [ sinh(\xi)cos(\xi) - cosh(\xi)sin(\xi) + j \left ( cosh(\xi) sin (\xi) + sinh(\xi) cos(\xi) \right ) \right ] \end{equation*}$

Relationship between complex solution and spatial solution

References

Parametri Dinamici UNI EN ISO 13786 – Prof. Marco Manzan

Luca Bonifazi – Comportamento termico dinamico di strutture opache