Uncertainty Analysis of Thermal Comfort Parameters

Abstract

International Standard ISO 7730:2005 defines thermal comfort as that condition of mind that expresses the degree of satisfaction with the thermal environment. Although this definition is inevitably subjective, the Standard gives formulae for two thermal comfort indices, predicted mean vote (PMV) and predicted percentage dissatisfied (PPD). The PMV formula is based on principles of heat balance and experimental data collected in a controlled climate chamber under steady-state conditions. The PPD formula depends only on PMV. Although these formulae are widely recognized and adopted, little has been done to establish measurement uncertainties associated with their use, bearing in mind that the formulae depend on measured values and tabulated values given to limited numerical accuracy. Knowledge of these uncertainties are invaluable when values provided by the formulae are used in making decisions in various health and civil engineering situations. This paper examines these formulae, giving a general mechanism for evaluating the uncertainties associated with values of the quantities on which the formulae depend. Further, consideration is given to the propagation of these uncertainties through the formulae to provide uncertainties associated with the values obtained for the indices. Current international guidance on uncertainty evaluation is utilized.

Appendix: Evaluating Sensitivity Coefficients

Consider the output quantity Y in a measurement model $Y = f(X_1, \ldots , X_N)$ having input quantities $X_1, \ldots , X_N$, where the measurement function f is given by a mathematical expression or computational algorithm.

A first-order sensitivity coefficient is $c_i = \partial f/\partial X_i$ evaluated at the best estimate ${\mathbf {x}}$ of ${\mathbf {X}}$. Then $u_i(y) = |c_i|u(x_i)$ is the corresponding first-order contribution to u(y), the standard uncertainty associated with the best estimate y of Y. The sensitivity coefficient $c_i$ can readily be obtained using the complex-step method [15], which in our opinion deserves much more prominence than it has, particularly when f is complicated. Rather than define the ith sensitivity coefficient as

$$\begin{aligned} c_i = \left. \frac{\partial f}{\partial X_i}\right| _{{\mathbf {X}} = {\mathbf {x}}}, \end{aligned}$$

it is defined by

$$\begin{aligned} c_i = \frac{1}{h}\mathfrak {I}f({\mathbf {x}} + \mathrm{i}h{\mathbf {e}}_i), \end{aligned}$$

where $\mathfrak {I}$ denotes “imaginary part,” h is a positive value much smaller than unity and ${\mathbf {e}}_i$ is the unit vector in the ith dimension.

This method is applicable when the real types in the software that implements the model can be replaced by complex types. Such types are supported by several popular computer languages.

The complex-step method is similar to finite differences but uses complex arithmetic to obtain first derivatives. It uses the Taylor expansion of a function f of a complex variable:

$$\begin{aligned} f(z + w) = \sum _{r=0}^{\infty } \frac{w^r}{r!} f^{(r)}(z), \end{aligned}$$

where z and w are complex. Setting $z = x$ and $w = \mathrm{i}h$ where x is real and h is real, positive and small, and taking real and imaginary parts,

$$\begin{aligned} \mathfrak {R}f(x \!+\! \mathrm{i}h)= & {} f(x) \!-\! \frac{h^2}{2} f'''(x) \!+\! O(h^4), \quad \mathfrak {I}f(x \!+\! \mathrm{i}h) \!=\! hf'(x) \!-\! \frac{h^3}{6} f'''(x) \!+\! O(h^5), \end{aligned}$$

from which

$$\begin{aligned} f(x) = \mathfrak {R}f(x + \mathrm{i}h), \qquad f'(x) = \frac{1}{h}\mathfrak {I}f(x + \mathrm{i}h), \end{aligned}$$

with truncation errors of order $h^2$. Unlike the use of a finite-difference formula for $f'(x)$, h is chosen to be very small with no concern about the loss of significant digits through subtraction cancellation. NPL routinely applies the method with $h = 10^{-100}$ [16], suitable for all but pathologically scaled problems. Also see references [17, 18].

A simple example of the use of the complex-step method is as follows. Consider the first derivative $f'(x)$ of the mathematical function,

$$\begin{aligned} f(x) = \frac{(1 + x)(1 + 2x)}{(1 + 3x)(1 + 4x)} \end{aligned}$$

evaluated at $x = 1$. A MATLAB implementation is

$$\begin{aligned} \mathtt{h }= & {} \mathtt{1e-16 };\\ \mathtt{x }= & {} \mathtt{1 + sqrt(-1)*h };\\ \mathtt{f }= & {} \mathtt{(1 + x)*(1 + 2*x)/((1 + 3*x)*(1 + 4*x)) };\\ \mathtt{dfdx }= & {} \mathtt{imag(f)/h }; \end{aligned}$$

The exact value of $f'(1)$ is $-23$/200, which the above code delivers correct to within one binary digit of full machine precision.