*3.2. Solution of PPD and Verification of Numerical Simulation* 3.2.1. Solution of *PPD*

The *PMV* index represents the feelings of most people in the same environment, but there are physiological differences between people, so the *PMV* index does not necessarily represent the feelings of all people. For this, Fanger [47] proposed the predicted percent dissatisfied (*PPD*) index to indicate the dissatisfaction rate of the population with the thermal environment, and gave the quantitative relationship between *PPD* and *PMV* by the probability analysis method. *PPD* has the following relationship with *PMV* [48]:

$$PPD = 100 - 95 \cdot \exp\left[ -0.2179 PMV^2 - 0.03353 PMV^4 \right] \tag{12}$$

Bring Equation (11) into Equation (12). The visual relationship between the *PPD* and indoor air temperature *ta* and floor surface temperature *ts* is shown in Figure 5.

**Figure 5.** Relationship between *PPD* and two temperature indexes.

It can be seen from Figure 5 that the relationship between *PPD* and two temperature variables (*ta* and *ts*) is also a group of parallel lines. When the floor surface temperature and indoor air temperature are extremely distributed (both very high or low), the *PPD* value is large. When the floor surface temperature and indoor air temperature are extremely distributed (both high or low), the *PPD* value is large. Only by maintaining the dynamic balance between *ta* and *ts* can the *PPD* value be minimized; that is, when the indoor air temperature is very high, a lower floor surface temperature is required, and when the indoor air temperature is low, a higher floor surface temperature is required. For an ideal *PPD* value (5~5.5%), *ta* and *ts* should be distributed in the ideal range of Figure 5, which satisfies the following inequality Equation (13):

$$\begin{aligned} &(-0.1936 \cdot t\_{\sf s} + 31.143 < t\_{\sf s} < -0.1955 \cdot t\_{\sf s} + 32.282\\ &t\_{\sf s} \in [18, 25 \, ^\circ \text{C}] \end{aligned} \tag{13}$$

3.2.2. Verification and Error Analysis

According to the usage habits of air conditioning, people generally set the indoor air temperature as an integer temperature. When the indoor air temperature is 26 ◦C, the *PPD* results of the numerical simulation of the present study are compared with a previous study [49]. The comparison is illustrated in Figure 6.

**Figure 6.** Results of the present study compared with a previous study [49].

A good agreement was achieved between the data of the previous study and present results, which showed the same trend of changes, and the maximum relative error was about 8.49%.

The mean absolute percentage error (*MAPE*) and the root mean square error (*RMSE*) [50] are employed to evaluate the accuracy of numerical simulation calculation models and boundary condition settings. The values of *MAPE* and *RMSE* were 6.64% and 0.60 respectively, which is acceptable.

This fully proves the rationality of the theoretical model and parameter settings used in the numerical simulations of the present study, and verifies the high reliability of the numerical results of this study. The numerical calculation model of the TRNSYS is verified.

#### *3.3. Correction Equation of PMV*

#### 3.3.1. Effect of Relative Humidity on *PMV*

Changes in the relative humidity also have an impact on human thermal comfort. This section discusses the influence of changes in independent variable *γ* on the *PMV* and corrects the calculation expression of *PMV* to obtain a calculation model containing independent variable *γ*.

Using the control variable method, fix the values of the value of *ta* and *ts*, and select the working conditions of *ta* = 28 ◦C, *ts* = 23 ◦C as condition I, *ta* = 28 ◦C, *ts* = 21 ◦C as condition II, *ta* = 27 ◦C, *ts* = 24 ◦C as condition III, *ta* = 27 ◦C, *ts* = 22 ◦C as condition IV, and *ta* = 26 ◦C, *ts* = 25 ◦C as condition V. For working conditions I~V, the indoor air relative humidity changes from 40% to 60% (in the summer, the relative indoor humidity of office buildings is generally between 40% and 60%), and a *PMV* value is output every 5% change in the relative humidity of the indoor air. The changes in the *PMV* are shown in Figure 7.

**Figure 7.** Influence of relative humidity changes on *PMV* under conditions I~V.

As can be seen from Figure 7, the *PMV* value increases with the increase in the relative humidity, which indicates that an increase in the relative humidity will increase people's warm feeling. This conclusion is consistent with the conclusion in [51], which verifies the reliability of numerical simulation from the side.

For each working condition in Figure 7, the changes in relative humidity had little impact on the *PMV*, and the maximum deviation (compared with the relative humidity at 50%) of the *PMV* under each working condition is less than 0.1.

It can be seen from Equation (12) that the *PMV* is the independent variable of *PPD*, and the change of the *PMV* is very small, so the change of *PPD* is also smaller.

#### 3.3.2. Complete Calculation Expression of *PMV*

In Figure 7, it can be found that the sample points of each working condition are approximately connected in a straight line, and the five straight lines are almost parallel to each other. This shows that the effect of the change in the relative humidity on each working condition is almost the same. A straight line or a formula can be used to represent the deviation of the *PMV* caused by a change in the relative humidity. When the indoor relative humidity changes, the offset value Δ of the *PMV* under each working condition is shown in Table 5. When the relative humidity deviation is the same, the average offset Δ of *PMV* is also given in Table 5.

**Table 5.** *PMV* deviation with relative humidity deviation (compared to 50%).


It can be found from Table 5 that, when the indoor relative humidity deviation is the same, the *PMV* deviation of each working condition is almost the same, and the average deviation value can be used to replace the actual deviation value. The average offset Δ can be used to correct the *PMV* when the relative humidity is not 50%. Through data fitting, the functional relationship between Δ and *γ* is found:

$$
\overline{\Delta} = 0.6944 \cdot (\gamma - 50\%) - 0.0031 \tag{14}
$$

Equation (14) is the correction formula for *PMV*, and the complete calculation formula of *PMV* is as follows:

It can be found from Table 5 that, when the indoor relative humidity deviation is the same, the *PMV* deviation of each working condition is almost the same, and the average deviation value can be used to replace the actual deviation value. The average offset Δ can be used to correct the *PMV* when the relative humidity is not 50%. Through data fitting, the functional relationship between Δ and *γ* is found:

$$
\overline{\Delta} = 0.6944 \cdot (\gamma - 50\%) - 0.0031 \tag{14}
$$

Equation (14) is the correction formula for *PMV*, and the complete calculation formula of *PMV* is as follows:

$$\begin{aligned} PMV &= \lg(t\_s, t\_a, \gamma) = \lg(t\_s, t\_a, 50\%) + \overline{\Delta} \\ &= 0.2826 t\_a + 5.5375 \times 10^{-2} t\_s - 8.9709 + 0.6944 \\ &\cdot (\gamma - 50\%) - 0.0031 \\ &= 0.2826 t\_a + 5.5375 \times 10^{-2} t\_s + 0.6944 \cdot \gamma - 9.3195 \\ t\_a \in [26, 29 \text{ } ^\circ \text{C}], \ t\_s \in [18, 25 \text{ } ^\circ \text{C}], \ \gamma \in [40\%, 60\%] \end{aligned} \tag{15}$$

where *ta* and *ts* participate in the calculation with ◦C as the unit.

For Equation (15), because the relative humidity has no unit, the indoor air temperature and the floor surface temperature are in ◦C, so the coefficient before γ can no longer be used for weight calculation; however, according to this coefficient, we can know that, for every 5% change in relative humidity, PMV will change by about 0.0347.

Equation (15) is the complete expression form of *g*(*ts*, *ta*, *γ*). PMV can be calculated according to the indoor air temperature *ta*, floor surface temperature *ts*, and indoor relative humidity *γ*. Among the three variables that affect PMV, air temperature is the main factor affecting PMV, floor surface temperature is a secondary factor affecting PMV, and relative humidity has little effect on PMV.

#### **4. Conclusions**

Some office buildings in northern China try to use radiant floors for cooling in the summer. Indoor thermal comfort is the key problem of a room using FRC technology, which was explored by the TRNSYS numerical simulation, control variable, and data fitting methods in this paper. The main conclusions are as follows.


A simplified calculation model of *PMV* for indoor thermal environment using floor radiant cooling was obtained. This simplified calculation model is suitable for changes in the indoor relative humidity, which contains three independent variables: the indoor air temperature *ta*, floor surface temperature *ts*, and indoor relative humidity *γ*. These three variables are the main factors affecting the indoor thermal environment. People can quickly calculate the *PMV* value based on this formula, or adjust one of the three variables with the *PMV* as an indicator to achieve the desired *PMV* value.

Additionally, the limitation of the adjusted temperature range of indoor air and floor surface caused by condensation on the floor surface has not been considered, which can be a research topic in the future.

**Author Contributions:** Conceptualization, L.Z. (Linhua Zhang); methodology, X.W.; software, X.W.; validation, T.M.; formal analysis, X.W.; investigation, T.M.; resources, L.Z. (Linhua Zhang); data curation, L.Z. (Lili Zhang); writing—original draft preparation, X.W.; writing—review and editing, W.Z.; visualization, X.W.; supervision, L.Z. (Linhua Zhang); project administration, W.Z.; funding acquisition, W.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Plan of Guidance and Cultivation for Young Innovative Talents of Shandong Provincial Colleges and Universities.

**Institutional Review Board Statement:** The study was conducted in accordance with the Declaration of Helsinki, and approved by the School of Thermal Engineering, Shandong Jianzhu University.

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** Data are not publicly available due to restrictions regarding the privacy of the participants.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**


#### **References**

