Investigations of hot water temperature changes at the pipe outflow

. In this paper a process of cold water withdrawing from hot water supply pipe systems without recirculation is considered. System of partial differential equations was used to describe the pipe and water temperature changes. An exact solution of a simplified form of the equations was obtained and validated experimentally. The exact solution was applied to calculate the hot water temperature changes at the pipe outflow. Calculations were done for typical pipe materials (PP, PE, Cu), different pipe diameters and lengths as well as for various water flow rates. It was shown that in order to obtain the required hot water temperature in the tap, there is necessary to withdrawn much more (even two times) water from the pipe in comparison to the pipe volume. The reason of such significant water wastes is a heat exchange between hot water flowing inside the pipe and the colder pipe walls. The results can be useful for optimal selection of hot water supply pipes as well as for making decision about applying of hot water recirculating systems.


Introduction
Domestic hot water in supply pipes will cool due to heat loss when it will be stagnant for quite some time. In large domestic hot water systems, where the distribution piping is extensive, it will take a relatively long time to remove this cold water from the pipes before the hot water can be available at the tap. This wastes both water and energy are often not acceptable. To overcome this problem, a water recirculation system can be used. Unfortunately some of these systems are very wasteful. According to the regulation [1] the maximum internal volume of the hot water supply pipe for which recirculation system is not required is 3 dm 3 . The aim of this paper is to investigate the effect of pipe type as well as an influence of flow rate on hot water temperature changes at the pipe outflow. In particular, we are interested in time needed to obtain a required outflow water temperature and mass of water withdrawn from the pipe before the accepted temperature is reached.

Theoretical model
Temperature changes of water (Tc) and pipe (Tr) are described in simplified form as [2]: with boundary conditions:  (2), a lumped thermal capacity model [3] for pipe and assumption of uniform radial temperature distribution in water are used. The thermal insulation of the pipe is perfect. Functions Nc, τc i τr were calculated as follows:  Initially in the insulated experimental pipe was cold water, i.e. the water taken from cold water supply pipe. The experimental run was started closing the recirculation hot water loop and opening the supply pipe hot water flow. During the hot water flow the outlet temperature was rising asymptotic until reaching the inlet hot water temperature. The whole temperature history was recorded with the 2 sec step.   The differences are slightly higher at larger water velocities and their explanation can be find in simplified form of the theoretical model. Applying simple time correction in form of (t + Δt) to the theoretical results it can be obtain a quite good agreement between experimental and corrected theoretical results. Finally, i.e. after "shifting of time axis", the discrepancies between dots and red lines shown in Figs. 3-5 were found as acceptable.

Effect of pipe type
A corrected theoretical model was applied for calculation of water outflow temperature changes for three standard commercial hot water supply pipes (Table 1). The pipe lengths Lr shown in Table 1 were obtained assuming that each pipe contains 3 dm 3 of water. According to the regulation [1], this is the maximum value. Symbols d1 and d2 mean inner and outer pipe diameters, respectively. The water velocities w were calculated for the flow rate 163 g/s in each pipe. The temperatures of cold and hot water were T0 = 20°C and T1 = 60°C. Pipe thermal capacity Qr was calculated as Qr = mr cr (T1-T0). Calculation results in form of outflow water temperature changes are presented in Fig. 6.

Time [s]
Cu PE PP Fig. 6. Water temperature changes at the pipe outflow for 3 different pipes (Table 1) of identical internal volume (3 dm 3 ), the same water flow rate (163 g/s), cold and hot temperatures 20°C and 60°C, respectively.
In Table 2 times t55 needed to obtain outflow water temperature of 55 o C, and masses m55 of water withdrawing from the pipe until the water outflow temperature reaches 55°C are given. Table 2. Times t55 and masses m55 for different pipes (pipes data: see Table 1). As one can see in Table 1, the PP pipe has the smallest internal surface Ar, the lowest heat transfer coefficient hc, and the largest heat capacity Qr. Therefore the intensity of heat exchange between water and PE pipe is smaller in comparison with the Cu and PE pipes. This is the most important cause of the discrepancies in temperature changes shown in Fig.  6. In Table 2 it is seen that for Cu pipe the smallest values of time t55 and mass m55 were obtained. For this pipe the mass m55 = 3,1 kg is only a little bit larger than the initial mass of cold water in the pipe. Also for Cu pipe, the time t55 = 19 s is only 1 s larger in comparison with time necessary to withdrawn the whole cold water from the pipe. For PP pipe, the m55 and t55 values are two time larger than for Cu pipe. For PE pipe, m55 and t55 reach intermediate values.

Effect of water velocity
PE pipe (pipe data: see Table 1) was selected to present the effect of water velocity on outflow water temperature changes. The results are shown in Fig. 7   As is seen, if water velocity increases, time necessary to reach an acceptable outflow temperature decreases and simultaneously mass of waste cold water increases.

Conclusions
The simply theoretical model was experimentally verified and introducing a time correction it was found useful for investigations of hot water temperature changes at the pipe outflow. The best hot water supply pipe should have small diameter and low thermal capacity value. Thermal insulation of the pipe ought to be as good as possible. An effect of the hot water flow rate on time needed to obtain a required outflow water temperature and the mass of cold water withdrawn from the pipe were investigated. It was found that with increasing of water velocity a decrease of time necessary to reach an acceptable outflow temperature and simultaneously an increase of losses of cold water are noticeable. The results ca be applied for appropriate selection of hot water supply pipes, i.e. length, diameter and material. They can be also useful for making decision about applying of hot water recirculating systems.