Selection of the Optimal Heat Carrier Velocity in Water Heating Networks
Abstract Generated abstract
The paper addresses the selection of economically optimal heat carrier velocity in water heat networks, criticizing the use of standardized or average specific head losses that may lead to excessive pipeline capital costs or pumping expenses. It develops a techno economic calculation in which annual costs per meter of pipeline include pumping energy, pipeline capital charges, pump station capital charges, and heat losses, then derives an explicit expression for the velocity that minimizes these costs. The resulting formula indicates that optimal velocity depends mainly on local economic and operating factors, such as electricity and heat prices, pump utilization hours, construction costs, loss coefficients, and maintenance deductions, rather than on network length or water flow rate. A numerical example suggests that velocities implied by prevailing design standards are often below economically advantageous values, especially for smaller diameters and regions with lower pump utilization.
Full Text
Selection of the Optimal Heat-Carrier Velocity in Water Heat Networks
E. N. Shadrin, N. A. Popov
(Presented by the scientific seminar of the Faculty of Thermal Power Engineering)
In connection with the large volume of construction of heat networks, the proper selection of the design parameters of heat networks, which determine the production costs of heat transport, is becoming increasingly important.
One of the principal quantities determining these costs is the specific linear pressure loss, or, equivalently, the corresponding heat-carrier velocity \(\omega\).
The method published in the literature for selecting the optimal specific head loss is rather complicated and presupposes the solution of this problem by successive approximation. On the other hand, the design standards for heat networks [3], while establishing an upper limit of the specific linear head loss for main heat pipelines of \(80\ \text{kg}/\text{m}^2\,\text{m}\), give no recommendations for its selection under local specific conditions. Therefore design organizations, lacking such recommendations, adopt in calculations average values of the specific linear head loss, usually within \(30\text{--}50\ \text{kg}/\text{m}^2\,\text{m}\), which corresponds to a heat-carrier velocity in pipelines of diameter \(200\text{--}1000\ \text{mm}\) from \(0.7\) to \(2.4\ \text{m/sec}\).
Such an arbitrary choice of water velocity may lead either to unjustified capital and metal expenditures in heat networks, or to excessive expenditures for pumping the heat carrier.
Therefore the optimal water velocity \(\omega_{\mathrm{v}}\) must be determined in each specific case from the condition of minimum calculated production costs for the delivery of a specified quantity of heat. In doing so, it is possible to take into account only those components of the total costs, referred to \(1\ \text{m}\) of heat-pipeline laying, that vary as a function of \(\omega_{\mathrm{v}}\). These components include the costs of pumping the heat carrier \(S_{\mathrm{per}}\), the costs associated with capital expenditures for laying the heat pipelines of the heat networks \(S_{\mathrm{t}}\), the pumping substation \(S_{\mathrm{n}}\), and the costs of heat losses \(S_{\mathrm{tp}}\).
Thus,
\[ S_{\mathrm{god}} = S_{\mathrm{per}} + S_{\mathrm{t}} + S_{\mathrm{n}} + S_{\mathrm{tp}} \quad \text{rub/year m}. \tag{1} \]
The optimal heat-carrier velocity is found from the condition:
\[ \frac{dS_{\mathrm{god}}}{d\omega_{\mathrm{v}}} = \frac{dS_{\mathrm{per}}}{d\omega_{\mathrm{v}}} + \frac{dS_{\mathrm{t}}}{d\omega_{\mathrm{v}}} + \frac{dS_{\mathrm{n}}}{d\omega_{\mathrm{v}}} + \frac{dS_{\mathrm{tp}}}{d\omega_{\mathrm{v}}}. \tag{2} \]
The component of the costs for pumping the heat carrier that varies with \(w_{\mathrm{v}}\) can be determined as
\[ S_{\mathrm{pump}}=\frac{3.6\cdot 10^{-3} cW\cdot H_{\mathrm{n}}A}{\eta_{\mathrm{n}}}\ \text{rub/year m}, \tag{3} \]
where
\(W\) is the quantity of network water being pumped, \(\text{m}^3/\text{s}\);
\(H_{\mathrm{n}}\) is the variable component of the head of the network pumps, \(\text{N}/\text{m}^2\);
\(\eta_{\mathrm{n}}\) is the efficiency of the network pumps;
\(c\) is the cost of electricity at the terminals of the thermal power plant, rub/kWh;
\(A\) is the number of hours of use of the installed capacity of the network pumps, h/year.
In formula (3), the variable component of the head of the network pumps \(H_{\mathrm{n}}\) consists of the head losses of the supply \(\Delta H_{\mathrm{p}}\) and return \(\Delta H_{0}\) pipelines.
These components will be found as
\[ H_{\mathrm{n}}=\Delta H_{\mathrm{p}}+\Delta H_{0} =\frac{\lambda\cdot \beta \rho_{\mathrm{t}}^{0.5} w_{\mathrm{v}}^{2.5}}{2W^{0.5}}\ \text{N}/\text{m}^3\text{m}. \tag{4} \]
Here \(\lambda\) is the friction coefficient;
\(\beta\) is the coefficient of local pressure losses;
\(\rho\) is the density of the heat carrier, \(\text{kg}/\text{m}^3\).
Substituting (4) into (3), we obtain
\[ S_{\mathrm{pump}}= \frac{3.6\cdot 10^{-3} cW\cdot A\lambda\beta\rho_{\mathrm{t}}^{0.5} w_{\mathrm{v}}^{2.5}}{2\eta_{\mathrm{n}}} = M_{1} w_{\mathrm{v}}^{2.5}\ \text{rub/year m}. \tag{5} \]
The cost of 1 m of heat pipelines as laid can be expressed as
\[ K_{\mathrm{t}}=a d_{y}\ \text{rub/m}. \]
where
\(a\) is a coefficient representing the cost of 1 m of heat pipeline as laid, per 1 m of pipeline diameter;
\(d_{y}\) is the nominal diameter of the heat pipeline, m.
According to data from the institute “Teploelektroproekt” [1], Fig. 1 shows a graph of the dependence of the cost of laying a two-pipe heat pipeline in nonwalk-through prefabricated concrete channels with gland compensators and with the supply and return pipelines insulated with mineral wool on the diameter. From the graph it is seen that the value
\[ a=270\ \text{rub}/\text{m}^2. \]
Fig. 1. Dependence of the cost of 1 m of heat pipelines on their diameters for two-pipe heat networks with gland compensators.
Then the costs depending on the capital expenditures in heat pipelines, taking into account the effectiveness of capital expenditures, per meter of installation of heat pipelines of two-pipe heat networks are determined as follows:
\[ S_{\mathrm{t}}= \frac{2\cdot 10^{-2}(u_{\mathrm{t}}+\delta)a\cdot W^{0.5}}{\pi^{0.5} w_{\mathrm{v}}^{0.5}} = \frac{M_{2}}{w_{\mathrm{v}}^{0.5}}\ \text{rub/m}, \tag{6} \]
where \(U_{\mathrm{t}}\) is the percentage of deductions for depreciation and current repair of heat pipelines;
\(\delta\) is the coefficient of effectiveness of capital expenditures.
The cost of pumps with installation and foundations may be taken as proportional to their power [2]
\[ K_{\mathrm{n}}=\kappa_{\mathrm{n}}N_{\mathrm{n}}=\kappa_{\mathrm{n}}\frac{W H_{\mathrm{n}}}{1000}\ \text{rubles}. \tag{7} \]
Here \(\kappa_{\mathrm{n}}\) is the cost of pumps per \(1\ \text{kW}\) of installed capacity, rubles/kW.
Then the production costs dependent on capital expenditures for network pumps, taking into account the effectiveness of capital expenditures, will be expressed as
\[ S_{\mathrm{n}}= \frac{10^{-2}(u_{\mathrm{n}}+\delta)\kappa_{\mathrm{n}}W^{0.5}\lambda\beta\rho\pi^{0.5}w_{\mathrm{v}}^{2.5}} {2\cdot 10^{3}} =M_{3}w_{\mathrm{v}}^{2.5}\ \text{rubles/year m}. \tag{8} \]
The costs associated with heat losses are
\[ S_{\mathrm{tp}}= \frac{4\pi q\beta'\Delta t_{\mathrm{av}}\,np\,W^{0.5}} {10^{6}\pi^{0.5}w_{\mathrm{v}}^{0.5}} = \frac{M_{4}}{w_{\mathrm{v}}^{0.5}}\ \text{rubles/year m}, \tag{9} \]
where \(q\) is the average heat flux from the heat carrier to the environment, referred to the surface of the pipes;
\(\Delta t_{\mathrm{av}}\) is the average annual temperature difference between the heat carrier and the medium surrounding the heat pipeline, \({}^{\circ}\mathrm{C}\);
\(n\) is the number of hours of operation of the heat network, hours/year;
\(\beta'\) is the coefficient of local heat losses;
\(p\) is the cost of lost heat, rubles/MJ.
If in equation (1), instead of the quantities entering into it, we substitute their values from expressions 5, 6, 8, 9, then we obtain
\[ S_{\mathrm{year}}=M_{5}w_{\mathrm{v}}^{2.5}+\frac{M_{6}}{w_{\mathrm{v}}^{0.5}}\ \text{rubles/year m}, \tag{10} \]
where
\[ M_{5}=M_{1}+M_{3};\qquad M_{6}=M_{2}+M_{4}. \]
To find the optimum velocity we take the first derivative \(\dfrac{dS_{\mathrm{year}}}{dw_{\mathrm{v}}}\) and set it equal to zero. Then, after several transformations, we obtain the expression for the optimum velocity of water in heat pipelines
\[ w_{\mathrm{v}}^{\mathrm{opt}}= \sqrt[3]{\frac{0.5M_{6}}{2.5M_{5}}}\ \text{m/sec}. \]
Substituting the values of \(M_{5}\) and \(M_{6}\), we obtain
\[ w_{\mathrm{v}}^{\mathrm{opt}}= \sqrt[3]{ \frac{ 2\cdot 10^{3}\left[0.01(u_{\mathrm{t}}+\delta)a+2\cdot 10^{-6}\pi q\beta'\Delta t_{\mathrm{av}}np\right] }{ 2.5\pi\lambda\beta\rho \left[ \dfrac{3.6\,cA}{\eta_{\mathrm{n}}} +0.01(u_{\mathrm{n}}+\delta)\kappa_{\mathrm{n}} \right] } }. \tag{11} \]
If in formula (11) one takes the mean value \(\rho=978\ \text{kg/m}^{3}\), \(\Delta t_{\mathrm{av}}=75^{\circ}\mathrm{C}\), \(n=8760\ \text{hours/year}\), then the formula assumes a simpler form:
\[ w_{\mathrm{v}}^{\mathrm{opt}}= \sqrt[3]{ \frac{ 0.26\left[0.01(u_{\mathrm{t}}+\delta)a+3.6\beta'qp\right] }{ \lambda\beta \left[ \dfrac{3.6\,cA}{\eta_{\mathrm{n}}} +0.01(u_{\mathrm{n}}+\delta)\kappa_{\mathrm{n}} \right] } }\ \text{m/sec}. \tag{12} \]
As is evident from formula (12), the water velocity in main heat pipelines depends neither on the amount of water being pumped nor on the length of the main heat pipelines, but is a function of such economic categories as the cost of laying 1 m of heat pipelines, the cost of electric power and heat, the cost of 1 kW of installed capacity of network pumps, the number of hours of utilization of the installed capacity of network pumps, the percentage deductions for depreciation and current repairs of heat pipelines and the pumping station, the coefficients of local hydraulic and heat losses, the friction coefficient, and the heat flow.
If we assume that the values \(u_{\mathrm{t}}\), \(u_{\mathrm{n}}\), and \(\delta\) at this stage remain constant, and that changes in the values \(k_{\mathrm{n}}\), \(\eta_{\mathrm{n}}\), \(\lambda\), \(\beta\), \(\beta'\), and \(q\) are insignificant, then the heat-carrier velocity in heat pipelines \(w_{\mathrm{v}}\) will basically depend on the number of hours of utilization of the installed capacity of the network pumps, and on the cost of electric power, heat, and laying the heat pipelines.
Fig. 2. Change in the optimal heat-carrier velocity as a function of the cost of electric power, the utilization factor of the installed capacity of network pumps, and the diameter of the heat pipeline.
In Fig. 2, by formula (12), for main heat pipelines of branched heat networks, the dependences of the optimal water velocity \(w_{\mathrm{v}}^{\mathrm{opt}}\) (shown in the graph by dashed lines) on the utilization factor of the maximum capacity of network pumps and on the cost of electric power are plotted for a particular example in which \(u_{\mathrm{t}} = 55\%\); \(u_{\mathrm{n}} = 8.50\%\); \(\delta = 12.5\%\); \(k_{\mathrm{n}} = 90\) rub/kW; \(\eta_{\mathrm{n}} = 0.7\); \(a = 270\) rub/m; \(p = 0.0007\) rub/MJ, and the values of the coefficients \(\lambda\), \(\beta\), \(\beta'\), and \(q\) are adopted according to the standards for designing heat networks as functions of the heat-pipeline diameter. The coefficient \(A\) for different climatic conditions is taken within the range of 500–5000 h/year; the cost of electric power on the busbars of a cogeneration plant is 0.001–0.0025 rub/MJ.
From the graph in Fig. 2 it can be seen that the water velocities in the main heat pipelines of water heat networks, adopted according to standardized specific linear head losses (shown in the graph by solid lines), are underestimated over almost the entire range of diameters in comparison with the economically advantageous velocities. This leads to overestimated pipeline diameters and, consequently, overestimated capital and metal expenditures for heat networks. This difference is greater the smaller the number of hours of utilization of the installed capacity of district-heating equipment and the cheaper the electric power generated at the cogeneration plant. This difference becomes especially noticeable for heat-pipeline diameters below 500 mm.
The optimal water velocities in transit mains will be still higher than in the main heat pipelines of branched heat networks.
Conclusions
- The water velocities selected in hydraulic calculations of main heat pipelines in heat networks in accordance with design standards are underestimated, and their selection should be based-
be based on techno-economic calculations that take specific local conditions into account.
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Since the optimal heat-carrier velocities are higher than those currently adopted, calculation of heat networks according to \(w_{\mathrm{v}}^{\mathrm{opt}}\) will lead to a reduction in capital and metal expenditures.
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In view of the diversity of climatic conditions, which lead to a large difference in the utilization factor of the installed capacity of district-heating equipment, identical water velocities in main heat pipelines in different regions of the country cannot be adopted.
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The water velocities in existing heat networks, provided they are maintained in normal condition, may, when it is necessary to increase their throughput capacity, be brought up to economically advantageous values.
References
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Consolidated indicators of the estimated cost of construction (in 1955 prices and standards), Part VII. External heat networks, VTPИ Publishing House, TEP, Moscow, 1956.
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E. N. Shadrin. Selection of the optimal water velocity in circulating water conduits, Proceedings of Higher Educational Institutions of the USSR—Power Engineering. Minsk, No. 5, 1963.
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Building Codes and Regulations, Part II, Section G, Chapter 10. Heat networks. Design standards. (SNiP, II-G-10-62), Moscow, 1964.