Steady StateTurbulent flow in the pipe. Medium temperature changes only with coordinate x. Pipe wall and insulation temperature change with x and r. Heat transfer between medium and pipe wall is so strong, that the inner pipe wall temperature equals medium temperature (α 1 => ∞, T=T1).Medium´s change of enthalpie from x to (x+dx) equals the radial heat transfer (qp·dx) in tube section dx:
The heat flow per unit length qp at x depends on the geometry (d, s1, s2), the operational thermal conductivities of pipe wall and insulation (λ1, λ2), the heat transfer between insulation and environment by radiation and convection (αr, α4) and the temperature difference between medium and environment (TT4): Ul : thermal transmittance per unit length Temperature differences (TT2), (T2T3) and (T3T4) can be derived from the laws for thermal conduction and convection. The heat flow per unit length qp stays constant in all 3 sections (pipe wall, insulation, environment), since neither pipe wall nor insulation pick up or release heat (temperature profile doesn´t change with time).
According to [DIN EN ISO 12241, pages 1719] the following set of formulas describes free convection with x, d, s [m], α4 [W/(m² K)]. Laminar flow prevails for horizontal piping, turbulent flow for vertical piping.
According to [DIN EN ISO 12241, pages 1719] the following set of formulas describes forced convection with v [m/s], d,s [m], α4 [W/(m² K)] :
According to [DIN EN ISO 12241, pages 1720] radation can be calculated as follows: ε: Emissivity [] σ: StefanBoltzmannConstant (radiation constant of the black body) 5.67 x 10^{8} [W/(m² K^{4})] T3, T4 [K]
Substituting equations (3), (4) and (5) into (2) results in a formula for the thermal transmittance per unit length: Pipe suspensions, valves and flanges enlarge the thermal transmittance per unit lenght. [DIN EN ISO 12241, pages 3537, 4041] and [VDI 2055, pages 3537, 150153] indicate the amount by which Ul has to be raised. If the first suspension is of no importance, heat loss of the suspensions for pipe systems with a span of 1 meter can be approximated by a surcharge of 15% (indoor installation) and 25% (outdoor installation). How those surcharges develop with increasing span shows the following table:
If you want to calculate the losses of all suspensions, you have to multiply the table values by 2/1=2 when calculating 2 suspensions, by 3/2=1.5 with 3 suspensions, by 4/3=1.333 with 4 suspensions, by 5/4=1.2 with 5 suspensionsand so on. Example: span 2[m], 10 suspensions, Outdoor installation: Supplementary value for plant related thermal bridges = 10/9 x 12.5 = 13,9% Thermal transmittance of a cylinder can be calculated as follows Substituting equation (2) into equation (1) leads to equation (7). Equation (7) describes the temperature profile of the medium along the pipe axis: Thermal transmittance Ul and surface temperature T3 are calculated at every node (x = 0, x = x1, x = x2). Dew formation on the pipe surface occurs if medium temperature is below ambient temperature and partial water vapor pressure equals saturation pressure at surface temperature. Vapour barrier layers (aluminum foil) minimize dew formation. Instationary StateWhen the flow stops the temperature profile in every tube section dx changes with time. Neglecting axial heat transfer and the heat capacity of pipe wall and insulation we can write:It is assumed that no phase change occurs in the medium. Thermal transmittance Ul and surface temperature T3 are calculated for the following nodes
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