Performance of heat transport systems: least square method generated correlations of non-dimensional variables

Abstract

Waste heat and renewable energy are increasingly being recognized as important sources of heat for district heating systems and for industrial clients, and as a means to reduce fossil fuel consumption. One of the most important challenges of using this heat is to transport it from the source to the load in an efficient manner. The objective of this paper is to establish explicit relations between the performance and the essential design variables of a system that includes a heat source, a heat transport sub-system, and a heat load. For this purpose we have used dimensional analysis, a factorial design of experiment methodology with the least square method and obtained linear correlations between two performance indicators (the system effectiveness and its exergy efficiency) and non-dimensional groups which combine the physical and operational characteristics of the system. It has also been shown that the economic desirability of the system as measured by the Internal Rate of Return increases linearly with the system effectiveness. A sensibility analysis has determined the non-dimensional groups which have the most important effect on each performance indicator. The obtained correlations have been applied to solve the following two design problems: (1) find the optimum values of design parameters such as the pipe insulation thickness and the mass flow rate of the heat transport fluid in order to maximize the exergy efficiency of the system and (2) find the optimum distribution of a fixed total thermal conductance in order to maximize the system effectiveness and/or its exergy efficiency.

Appendix

The dimensional equations modeling the system are

$$ \mathrm{Q}\dot {}_{\mathrm{h}}={\dot{\mathrm{m}}}_{\mathrm{h}}{\mathrm{C}}_{\mathrm{ph}}\left({\mathrm{T}}_{\mathrm{h},\mathrm{in}}-{\mathrm{T}}_{\mathrm{h},\mathrm{out}}\right)={\dot{\mathrm{m}}}_{\mathrm{t}}{\mathrm{C}}_{\mathrm{pt}}\left({\mathrm{T}}_1-{\mathrm{T}}_4\right) $$

(29)

$$ {\dot{\mathrm{m}}}_{\mathrm{h}}{\mathrm{C}}_{\mathrm{ph}}\left({\mathrm{T}}_{\mathrm{h},\mathrm{in}}-{\mathrm{T}}_{\mathrm{h},\mathrm{out}}\right)={\mathrm{UA}}_{\mathrm{h}}\left[\left({\mathrm{T}}_{\mathrm{h},\mathrm{in}}-{\mathrm{T}}_1\right)-\left(\ {\mathrm{T}}_{\mathrm{h},\mathrm{out}}-{\mathrm{T}}_4\right)\right]/\ln \left[\left({\mathrm{T}}_{\mathrm{h},\mathrm{in}}-{\mathrm{T}}_1\right)/\left(\ {\mathrm{T}}_{\mathrm{h},\mathrm{out}}-{\mathrm{T}}_4\right)\right] $$

(30)

$$ {\dot{\mathrm{m}}}_{\mathrm{t}}{\mathrm{C}}_{\mathrm{pt}}\left({\mathrm{T}}_1-{\mathrm{T}}_2\right)\approx \left[\left({\mathrm{T}}_1+{\mathrm{T}}_2\right)/2-{\mathrm{T}}_{\mathrm{env}}\right]/\left({\mathrm{R}}_{\mathrm{m}}+{\mathrm{R}}_{\mathrm{ins}}\right) $$

(31)

$$ {\mathrm{R}}_{\mathrm{m}}=\ln \left({\mathrm{D}}_2/{\mathrm{D}}_1\right)/\left({2\uppi \mathrm{k}}_{\mathrm{m}}\mathrm{L}\right) $$

(32)

$$ {\mathrm{R}}_{\mathrm{ins}}=\ln \left({\mathrm{D}}_3/{\mathrm{D}}_2\right)/\left({2\uppi \mathrm{k}}_{\mathrm{ins}}\mathrm{L}\right) $$

(33)

$$ \mathrm{Q}\dot {}_{\mathrm{c}}={\dot{\mathrm{m}}}_{\mathrm{c}}{\mathrm{C}}_{\mathrm{pc}}\left({\mathrm{T}}_{\mathrm{c},\mathrm{out}}-{\mathrm{T}}_{\mathrm{c},\mathrm{in}}\right)={\dot{\mathrm{m}}}_{\mathrm{t}}{\mathrm{C}}_{\mathrm{pt}}\left({\mathrm{T}}_2-{\mathrm{T}}_3\right) $$

(34)

$$ {\dot{\mathrm{m}}}_{\mathrm{t}}{\mathrm{C}}_{\mathrm{pt}}\left({\mathrm{T}}_2-{\mathrm{T}}_3\right)={\mathrm{UA}}_{\mathrm{c}}\left[\left({\mathrm{T}}_2-{\mathrm{T}}_{\mathrm{c},\mathrm{out}}\right)-\left({\mathrm{T}}_3-{\mathrm{T}}_{\mathrm{c},\mathrm{in}}\right)\right]/\ln \left[\left({\mathrm{T}}_2-{\mathrm{T}}_{\mathrm{c},\mathrm{out}}\right)/\left({\mathrm{T}}_3-{\mathrm{T}}_{\mathrm{c},\mathrm{in}}\right)\right] $$

(35)

$$ {\dot{\mathrm{m}}}_{\mathrm{t}}{\mathrm{C}}_{\mathrm{pt}}\left({\mathrm{T}}_3-{\mathrm{T}}_4\right)\approx \left[\left({\mathrm{T}}_3+{\mathrm{T}}_4\right)/2-{\mathrm{T}}_{\mathrm{env}}\right]/\left({\mathrm{R}}_{\mathrm{m}}+{\mathrm{R}}_{\mathrm{ins}}\right) $$

(36)

These ten equations involve 23 dimensional variables; 13 can be classified as input parameters (ṁ_hC_ph, ṁ_tC_pt, ṁ_cC_pc, UA_h, UA_c, k_mL, k_insL, D₃, D₂, D₁, T_h,in, T_c,in, T_env) and 10 as dependent quantities (Q̇_h, Q̇_c, R_m, R_ins, T₁, T₂, T₃, T₄, T_c,out, T_h,out). The latter quantities can be calculated for any combination of the 13 input parameters by solving the above system of ten algebraic equations. Among the dependent quantities the heat transfer rate Q̇_c is of particular interest since it represents the useful effect of the system. The relation between Q̇_c and the input parameters is

$$ \mathrm{Q}\dot {}_{\mathrm{c}}=\mathrm{f}\ \left({\dot{\mathrm{m}}}_{\mathrm{h}}{\mathrm{C}}_{\mathrm{ph}},{\dot{\mathrm{m}}}_{\mathrm{t}}{\mathrm{C}}_{\mathrm{pt}},{\dot{\mathrm{m}}}_{\mathrm{c}}{\mathrm{C}}_{\mathrm{pc}},{\mathrm{UA}}_{\mathrm{h}},{\mathrm{UA}}_{\mathrm{c}},{\mathrm{k}}_{\mathrm{m}}\mathrm{L},{\mathrm{k}}_{\mathrm{ins}}\mathrm{L},{\mathrm{D}}_{3,}\ {\mathrm{D}}_2,{\mathrm{D}}_1,{\mathrm{T}}_{\mathrm{h},\mathrm{in}},{\mathrm{T}}_{\mathrm{c},\mathrm{in}},{\mathrm{T}}_{\mathrm{env}}\right) $$

(37)