Research on heating performance of direct-condensation thermal storage aluminum radiant heating panel under multiple working conditions

doi:10.11949/0438-1157.20240958

Abstract

Abstract:

A new type of aluminum, fanless, built-in heat storage material direct condensing radiant panel heat exchanger (AHE) is proposed, which can be combined with an air source heat pump system for indoor winter heating in buildings. Based on the heat transfer mechanism of the heat storage type direct condensation radiant panel and considering the influence of refrigerant flow on heat transfer performance, a flow heat transfer mathematical model suitable for AHE was established, and the accuracy of the model was verified through experiments. The results showed that the numerical simulation results of heat dissipation, pressure drop, and heat exchanger surface temperature had an average deviation of less than 5% from the experimental values. Numerical simulations were conducted on the thermal performance of AHE under 168 working conditions, and the results showed that increasing the condensation temperature and refrigerant flow rate helps to improve the heat transfer intensity, while increasing the condensation temperature is beneficial for reducing flow losses. In 168 working conditions, the average temperature difference between AHE refrigerant and surface temperature was 9.7℃, and the maximum temperature difference between adjacent structural layers of AHE was 6.3℃ between the copper tube and the water layer. Finally, a heat dissipation characteristic formula suitable for AHE thermal performance prediction was proposed, providing a technical basis for performance analysis and optimization of radiant heating panel systems.

Key words: radiation, numerical simulation, thermal storage materials, air source heat pump, formula for heat dissipation characteristics, optimization

摘要：

提出了一种新型铝制、无风机、内置蓄热材料的直接冷凝式辐射板换热器（aluminum radiant heating panel exchanger，AHE），可与空气源热泵系统结合进行建筑室内冬季供暖。基于蓄热型直凝式辐射板的传热机理并考虑制冷剂流动对换热性能的影响，建立了适合AHE的流动换热模型，并通过实验验证了该模型的准确性。结果表明，散热量、压降、换热器表面温度的数值模拟结果与实验值平均偏差小于5%。针对168组运行工况下AHE的热性能进行了数值模拟，结果表明冷凝温度和制冷剂流量的提升有助于提高换热强度，同时冷凝温度的升高有利于降低流动损失。在168组工况中，AHE制冷剂和表面温度的平均温差为9.7℃，AHE各相邻结构层最大温差在铜管和水层间，为6.3℃。最后提出了适用于AHE热性能预测的散热量特征公式，为辐射板供暖系统性能分析和优化提供了技术基础。

关键词: 辐射, 数值模拟, 蓄热材料, 空气源热泵, 散热特征公式, 优化

CLC Number:

TK 519

Chengcheng XU, Suola SHAO, Wenjian WEI, Xu ZHENG. Research on heating performance of direct-condensation thermal storage aluminum radiant heating panel under multiple working conditions[J]. CIESC Journal, 2025, 76(4): 1545-1558.

许成城, 邵索拉, 魏文建, 郑旭. 多工况下直凝式蓄热型铝制辐射板换热器供暖性能研究[J]. 化工学报, 2025, 76(4): 1545-1558.

Figures/Tables 14

Table 1 Metal thermal strength of different materials

型材	金属热强度范围/(W/(kg·℃))
钢（柱型）	0.6~1.3
钢（板型）	0.9~1.4
钢（对流型）	0.5~0.9
铜、铝及铜铝复合（柱翼型）	1.8~3.9
铜、铝及铜铝复合（对流型）	0.8~3.0
灰铸铁	0.3~0.45

Fig.1 Structure of the AHE

Fig.2 Testing bench and measuring points layout for the air source heat pump system with AHE

Fig.3 Heat transfer mechanism of the AHE

Table 2 The correlations of the heat transfer and pressure drop in the AHE model

类别	关联式	适用范围
Q_re	$Q_{r e} (i) = G_{r e} (i) (h_{r e - i n} (i) - h_{r e - o u t} (i))$	制冷剂热量
Q_conv,re-co	$Q_{c o n v, r e - c o} (i) = A_{c o - i n} (i) α_{c o n v, r e - c o} (i) (\frac{t_{r e - i n} (i) + t_{r e - o u t} (i)}{2} - t_{c o - i n} (i))$	制冷剂与铜管对流换热
	$N u_{c o n v, r e - c o} (i) = \frac{(f_{r e - g} (i) / 8) R e_{r e - g} (i) P r_{r e - g} (i)}{1 + 12.7 \sqrt[]{\frac{f_{r e - g} (i)}{8}} (P r_{r e - g} {(i)}^{\frac{2}{3}} - 1)} (1 + {\frac{d_{c o - i n}}{Δ l}}^{2 / 3}) {(\frac{T_{r e (i)}}{T_{c o - i n} (i)})}^{0.45}$ $f_{r e - g} (i) = (1.82 l g R e_{r e - g} {(i) - 1.64)}^{- 2}, R e_{r e - g} (i) = \frac{G_{r e} (i) d_{c o - i n}}{μ_{r e - g} (i)}$	单相过热区^[20] Re_re=2300~10⁵ Pr_re=0.6~10⁵ $\frac{T_{r e}}{T_{c o}}$ =0.5~1.5
	$N u_{c o n v, r e - c o} (i) = \frac{(f_{r e - l} (i) / 8) R e_{r e - l} (i) P r_{r e - l} (i)}{1 + 12.7 \sqrt[]{\frac{f_{r e - l} (i)}{8}} (P r_{r e - l} {(i)}^{\frac{2}{3}} - 1)} (1 + {\frac{d_{c o - i n}}{Δ l}}^{2 / 3}) {(\frac{P r_{r e (i)}}{P r_{c o - i n} (i)})}^{0.11}$ $f_{r e - l} (i) = [1.82 l g R e_{r e - l} {(i) - 1.64]}^{- 2}, R e_{r e - l} (i) = \frac{G_{r e} (i) d_{c o - i n}}{μ_{r e - l} (i)}$	单相过冷区^[20] Re_re=2300~10⁵ Pr_re=0.6~10⁵, $\frac{P r_{r e}}{P r_{c o - i n}}$ =0.05~20
	$N u_{c o n v, r e - c o} (i) = 0.023 R e_{r e} {(i)}^{0.8} P r_{r e} {(i)}^{0.3}$	单相过冷/热^[21] Re_re>10⁵, Pr_re=0.7~160
	$\begin{matrix} N u_{c o n v, r e - c o} (i) = [{(0.31 R e_{r e - l s} {(i)}^{- 1.32} + \frac{R e_{r e - l s} {(i)}^{2.4} P r_{r e - l} {(i)}^{3.9}}{2.37 \times 10^{14}})}^{1 / 3} + \\ {\frac{P r_{r e - l} {(i)}^{1.3}}{771.6} \times \frac{0.252 μ_{r e - l} {(i)}^{1.177} μ_{r e - g} {(i)}^{0.156}}{d_{c o - i n}^{2} g^{\frac{2}{3}} ρ_{r e - l} {(i)}^{0.553} ρ_{r e - g} {(i)}^{0.78}} (R e_{r e - l} (i) - R e_{r e - l s} {(i))}^{1.4} R e_{r e - l s} {(i)}^{0.4}]}^{1 / 2} \end{matrix}$ $R e_{r e - l s} (i) = \frac{G_{r e} (i) \times (1 - x_{r e} (i)) d_{c o - i n}}{μ_{r e - l} (i)}$	两相区^[22]
Q_cond,co-wa	$Q_{c o n d, c o - w a} (i) = Δ l \frac{t_{c o} (i) - t_{w a} (i)}{\frac{1}{2 π λ_{c o}} l n \frac{d_{c o - o u t}}{d_{c o - i n}} + \frac{1}{2 π λ_{w a}} l n \frac{(d_{a p - i n} + d_{c o - o u t}) / 2}{d_{c o - o u t}}}$	铜管与水层导热^[23]
Q_cond,wa-ap	$Q_{c o n d, w a - a p} (i) = Δ l \frac{t_{w a} (i) - t_{a p} (i)}{\frac{1}{2 π λ_{w a}} l n \frac{d_{a p - i n}}{(d_{a p - i n} + d_{c o - o u t}) / 2} + \frac{1}{2 π λ_{a p}} l n \frac{d_{a p - o u t}}{d_{a p - i n}}}$	水层与壳体导热^[23]
Q_rad,ap-bu	$Q_{r a d, a p - b u} (i) = \{\begin{array}{l} 5.67 \times 10^{- 8} A_{a p} [(t_{a p} {(i) + 273.15)}^{4} - (T_{A U S T} {+ 273.15)}^{4}] 前换热柱 \\ 0 后换热柱 \end{array}$ $T_{A U S T} = \frac{\sum_{1}^{6} A_{i} T_{i}}{\sum_{1}^{6} A_{i}}$	壳体与空气辐射换热^[23]
Q_{conv, ap-nofi-ai}	$Q_{c o n v, a p - n o f i - a i} (i) = A_{a p - n o f i} (i) α_{c o n v, a p - a i} (i) (t_{a p} (i) - t_{a i} (i))$ $A_{a p - n o f i} = \{\begin{matrix} Δ l L_{a p} & 前换热柱 \\ 0 & 后换热柱 \end{matrix}$ $α_{c o n v, a p - a i} (i) = \frac{N u_{c o n v, a p - a i} (i) k_{a i} (i)}{Δ l_{}}, N u_{c o n v, a p - a i} (i) = {\{0.825 + \frac{0.387 R a_{H} {(i)}^{1 / 6}}{[1 + (0.492 / P r {(i))}^{9 / 16}]^{8 / 27}}\}}^{2}$ $R a_{H} (i) = P r (i) \frac{g β (i) Δ l θ {(i)}^{3}}{υ {(i)}^{2}}, θ (i) = t_{a p} (i) - t_{a i} (i), β (i) = 1 / (t_{a i - f} (i) + 273.15)$ $t_{a i - f} (i) = \frac{t_{a i} (i) + t_{a p} (i)}{2}$	不含肋片的壳体与空气对流换热^[24]
Q_{conv, ap-fi-ai}	$Q_{c o n v, a p - f i - a i} (i) = α_{c o n v, a p - f i - a i} s_{f i} (\frac{t_{a p} (i) + t_{f i - o u t} (i)}{2} - t_{a i})$	含平行组合肋片的壳体与空气对流换热^[25]
Q_{conv, ap-fi-ai}	$s_{f i} = Δ l W_{f i} + 2 n_{f i} L_{f i} Δ l, α_{c o n v, a p - f i - a i} = N u_{c o n v, a p - f i - a i} \frac{λ_{f i}}{\sqrt[]{s_{f i}}}$ $N u_{c o n v, a p - f i - a i} = N u_{\sqrt[]{s_{f i}}}^{\infty} + f (P r) G_{\sqrt[]{s_{f i}}} R {a_{\sqrt[]{s_{f i}}}}^{1 / 4}$ $N u_{\sqrt[]{s_{f i}}}^{\infty} = \frac{3.192 + 1.868 {(\frac{δ_{a p} + L_{f i}}{Δ l})}^{0.76}}{\sqrt[]{1 + 1.189 \frac{(δ_{a p} + L_{f i})}{Δ l}}}$ $G_{\sqrt[]{s_{f i}}} = 1.0904 {\{\frac{Δ l (n_{f i} L_{f i} + δ_{f i} + W_{f i})^{2}}{[n_{f i} δ_{f i} L_{f i} + δ_{f i} W_{f i} + Δ l (n_{f i} L_{f i} + δ_{f i} + W_{f i} {)]}^{3 / 2}}\}}^{1 / 4}$ $R a_{\sqrt[]{s_{f i}}} = P r \frac{g β [\frac{t_{a p} (i) + t_{f i - o u t} (i)}{2} - t_{a i}] {(\sqrt[]{s_{f i}})}^{3}}{v^{2}}$ $β = 1 / [(\frac{t_{a p} (i) + t_{f i - o u t} (i)}{2} + t_{a i} + 273.15) / 2], f (P r) = 0.67 / [1 + {(0.5 / P r)}^{9 / 16}]^{4 / 9}$	含平行组合肋片的壳体与空气对流换热^[25]
Q_{conv, ap(fi)-ai}	$Q_{c o n v, a p (f i) - a i} (i) = A_{a p (f i)} (i) α_{c o n v, a p - a i} (i) (t_{a p} (i) - t_{a i} (i))$ $A_{a p (f i)} = \{\begin{matrix} Δ l L_{a p} - n_{f i} Δ l δ_{f i} & f o r f r o n t g r o u p s \\ 2 Δ l L_{a p} - 2 n_{f i} Δ l δ_{f i} & f o r r e a r g r o u p s \end{matrix}$	含有肋片的壳体平板部分与空气的对流换热
Q_{cond, ap-fi}	$Q_{c o n d, a p - f i} (i) = A_{f i} λ_{f i} \frac{(t_{a p} (i) - t_{f i - o u t} (i))}{δ_{f i}}$ $A_{f i} = \{\begin{matrix} n_{f i} Δ l δ_{f i} & f o r f r o n t g r o u p s \\ 2 n_{f i} Δ l δ_{f i} & f o r r e a r g r o u p s \end{matrix}$	壳体与肋片的导热
ΔP_m	$Δ P_{m} (i) = G_{r e}^{2} {[\frac{x_{r e} {(i)}^{2}}{ξ ρ_{r e - g} (i)} + \frac{(1 - x_{r e} {(i))}^{2}}{(1 - ξ) ρ_{r e - l} (i)}]}_{o u t} - G_{r e}^{2} {[\frac{x_{r e} {(i)}^{2}}{ξ ρ_{r e - g} (i)} + \frac{(1 - x_{r e} {(i))}^{2}}{(1 - ξ) ρ_{r e - l} (i)}]}_{i n}$ $ξ = 1 - \frac{2 (1 - x_{r e} {(i))}^{2}}{1 - 2 x_{r e} (i) + {[1 + 4 x_{r e} (i) (1 - x_{r e} (i)) (\frac{ρ_{r e - l} (i)}{ρ_{r e - g} (i)} - 1)]}^{0.5}}$	动量压降^[26]
ΔP_f	$Δ P_{f} (i) = - \frac{2 f (i) {G_{r e}}^{2} Δ l}{d_{c o - i n} ρ_{r e} (i)}$ , $f (i) = 0.0046 R e_{r e} {(i)}^{- 0.2}$	单相流流体摩擦压降^[27]
ΔP_f	$Δ P_{f} (i) = Δ P_{f - l} (i) ϕ {(i)}^{2}$ $ϕ {(i)}^{2} = 1 + \frac{C}{ψ} + \frac{1}{ψ^{2}}$ , $ψ^{2} = \frac{{(\frac{d P_{f}}{d z})}_{l}}{{(\frac{d P_{f}}{d z})}_{g}}$ ${(\frac{d P_{f}}{d z})}_{l} = \frac{2 f {(i)}_{l} {G_{r e}}^{2} (1 - x_{r e} {(i))}^{2}}{d_{c o - i n} ρ_{r e - l} (i)} {(\frac{d P_{f}}{d z})}_{g} = \frac{2 f {(i)}_{g} {G_{r e}}^{2} x_{r e} {(i)}^{2}}{d_{c o - i n} ρ_{r e - g} (i)}$ $C = \{\binom{\binom{0.39 R e_{r e - l}^{0.03} L a_{r e - g}^{0.1} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.35} R e_{r e - l} \geq 2000, R e_{r e - g} \geq 2000}{8.7 \times 10^{- 4} R e_{r e - l}^{0.17} L a_{r e - g}^{0.5} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.14} R e_{r e - l} \geq 2000, R e_{r e - g} < 2000}}{\binom{0.0015 R e_{r e - l}^{0.59} L a_{r e - g}^{0.19} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.36} R e_{r e - l} < 2000, R e_{r e - g} \geq 2000}{3.5 \times 10^{- 5} R e_{r e - l}^{0.44} L a_{r e - g}^{0.5} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.48} R e_{r e - l} < 2000, R e_{r e - g} < 2000}}$ $L a_{r e - g} = \frac{R {e_{r e - g}}^{2}}{W e_{r e - g}}, R e_{r e - g} = \frac{G_{r e} d_{c o - i n}}{μ_{r e - g}}, R e_{r e - l} = \frac{G_{r e} d_{c o - i n}}{μ_{r e - l}}, W e_{r e - g} = \frac{{G_{r e}}^{2} d_{c o - i n}}{ρ_{r e - g}}$	两相流流体摩擦压降^[28]

Table 2 The correlations of the heat transfer and pressure drop in the AHE model

类别	关联式	适用范围
Q_re	$Q_{r e} (i) = G_{r e} (i) (h_{r e - i n} (i) - h_{r e - o u t} (i))$	制冷剂热量
Q_conv,re-co	$Q_{c o n v, r e - c o} (i) = A_{c o - i n} (i) α_{c o n v, r e - c o} (i) (\frac{t_{r e - i n} (i) + t_{r e - o u t} (i)}{2} - t_{c o - i n} (i))$	制冷剂与铜管对流换热
	$N u_{c o n v, r e - c o} (i) = \frac{(f_{r e - g} (i) / 8) R e_{r e - g} (i) P r_{r e - g} (i)}{1 + 12.7 \sqrt[]{\frac{f_{r e - g} (i)}{8}} (P r_{r e - g} {(i)}^{\frac{2}{3}} - 1)} (1 + {\frac{d_{c o - i n}}{Δ l}}^{2 / 3}) {(\frac{T_{r e (i)}}{T_{c o - i n} (i)})}^{0.45}$ $f_{r e - g} (i) = (1.82 l g R e_{r e - g} {(i) - 1.64)}^{- 2}, R e_{r e - g} (i) = \frac{G_{r e} (i) d_{c o - i n}}{μ_{r e - g} (i)}$	单相过热区^[20] Re_re=2300~10⁵ Pr_re=0.6~10⁵ $\frac{T_{r e}}{T_{c o}}$ =0.5~1.5
	$N u_{c o n v, r e - c o} (i) = \frac{(f_{r e - l} (i) / 8) R e_{r e - l} (i) P r_{r e - l} (i)}{1 + 12.7 \sqrt[]{\frac{f_{r e - l} (i)}{8}} (P r_{r e - l} {(i)}^{\frac{2}{3}} - 1)} (1 + {\frac{d_{c o - i n}}{Δ l}}^{2 / 3}) {(\frac{P r_{r e (i)}}{P r_{c o - i n} (i)})}^{0.11}$ $f_{r e - l} (i) = [1.82 l g R e_{r e - l} {(i) - 1.64]}^{- 2}, R e_{r e - l} (i) = \frac{G_{r e} (i) d_{c o - i n}}{μ_{r e - l} (i)}$	单相过冷区^[20] Re_re=2300~10⁵ Pr_re=0.6~10⁵, $\frac{P r_{r e}}{P r_{c o - i n}}$ =0.05~20
	$N u_{c o n v, r e - c o} (i) = 0.023 R e_{r e} {(i)}^{0.8} P r_{r e} {(i)}^{0.3}$	单相过冷/热^[21] Re_re>10⁵, Pr_re=0.7~160
	$\begin{matrix} N u_{c o n v, r e - c o} (i) = [{(0.31 R e_{r e - l s} {(i)}^{- 1.32} + \frac{R e_{r e - l s} {(i)}^{2.4} P r_{r e - l} {(i)}^{3.9}}{2.37 \times 10^{14}})}^{1 / 3} + \\ {\frac{P r_{r e - l} {(i)}^{1.3}}{771.6} \times \frac{0.252 μ_{r e - l} {(i)}^{1.177} μ_{r e - g} {(i)}^{0.156}}{d_{c o - i n}^{2} g^{\frac{2}{3}} ρ_{r e - l} {(i)}^{0.553} ρ_{r e - g} {(i)}^{0.78}} (R e_{r e - l} (i) - R e_{r e - l s} {(i))}^{1.4} R e_{r e - l s} {(i)}^{0.4}]}^{1 / 2} \end{matrix}$ $R e_{r e - l s} (i) = \frac{G_{r e} (i) \times (1 - x_{r e} (i)) d_{c o - i n}}{μ_{r e - l} (i)}$	两相区^[22]
Q_cond,co-wa	$Q_{c o n d, c o - w a} (i) = Δ l \frac{t_{c o} (i) - t_{w a} (i)}{\frac{1}{2 π λ_{c o}} l n \frac{d_{c o - o u t}}{d_{c o - i n}} + \frac{1}{2 π λ_{w a}} l n \frac{(d_{a p - i n} + d_{c o - o u t}) / 2}{d_{c o - o u t}}}$	铜管与水层导热^[23]
Q_cond,wa-ap	$Q_{c o n d, w a - a p} (i) = Δ l \frac{t_{w a} (i) - t_{a p} (i)}{\frac{1}{2 π λ_{w a}} l n \frac{d_{a p - i n}}{(d_{a p - i n} + d_{c o - o u t}) / 2} + \frac{1}{2 π λ_{a p}} l n \frac{d_{a p - o u t}}{d_{a p - i n}}}$	水层与壳体导热^[23]
Q_rad,ap-bu	$Q_{r a d, a p - b u} (i) = \{\begin{array}{l} 5.67 \times 10^{- 8} A_{a p} [(t_{a p} {(i) + 273.15)}^{4} - (T_{A U S T} {+ 273.15)}^{4}] 前换热柱 \\ 0 后换热柱 \end{array}$ $T_{A U S T} = \frac{\sum_{1}^{6} A_{i} T_{i}}{\sum_{1}^{6} A_{i}}$	壳体与空气辐射换热^[23]
Q_{conv, ap-nofi-ai}	$Q_{c o n v, a p - n o f i - a i} (i) = A_{a p - n o f i} (i) α_{c o n v, a p - a i} (i) (t_{a p} (i) - t_{a i} (i))$ $A_{a p - n o f i} = \{\begin{matrix} Δ l L_{a p} & 前换热柱 \\ 0 & 后换热柱 \end{matrix}$ $α_{c o n v, a p - a i} (i) = \frac{N u_{c o n v, a p - a i} (i) k_{a i} (i)}{Δ l_{}}, N u_{c o n v, a p - a i} (i) = {\{0.825 + \frac{0.387 R a_{H} {(i)}^{1 / 6}}{[1 + (0.492 / P r {(i))}^{9 / 16}]^{8 / 27}}\}}^{2}$ $R a_{H} (i) = P r (i) \frac{g β (i) Δ l θ {(i)}^{3}}{υ {(i)}^{2}}, θ (i) = t_{a p} (i) - t_{a i} (i), β (i) = 1 / (t_{a i - f} (i) + 273.15)$ $t_{a i - f} (i) = \frac{t_{a i} (i) + t_{a p} (i)}{2}$	不含肋片的壳体与空气对流换热^[24]
Q_{conv, ap-fi-ai}	$Q_{c o n v, a p - f i - a i} (i) = α_{c o n v, a p - f i - a i} s_{f i} (\frac{t_{a p} (i) + t_{f i - o u t} (i)}{2} - t_{a i})$	含平行组合肋片的壳体与空气对流换热^[25]
Q_{conv, ap-fi-ai}	$s_{f i} = Δ l W_{f i} + 2 n_{f i} L_{f i} Δ l, α_{c o n v, a p - f i - a i} = N u_{c o n v, a p - f i - a i} \frac{λ_{f i}}{\sqrt[]{s_{f i}}}$ $N u_{c o n v, a p - f i - a i} = N u_{\sqrt[]{s_{f i}}}^{\infty} + f (P r) G_{\sqrt[]{s_{f i}}} R {a_{\sqrt[]{s_{f i}}}}^{1 / 4}$ $N u_{\sqrt[]{s_{f i}}}^{\infty} = \frac{3.192 + 1.868 {(\frac{δ_{a p} + L_{f i}}{Δ l})}^{0.76}}{\sqrt[]{1 + 1.189 \frac{(δ_{a p} + L_{f i})}{Δ l}}}$ $G_{\sqrt[]{s_{f i}}} = 1.0904 {\{\frac{Δ l (n_{f i} L_{f i} + δ_{f i} + W_{f i})^{2}}{[n_{f i} δ_{f i} L_{f i} + δ_{f i} W_{f i} + Δ l (n_{f i} L_{f i} + δ_{f i} + W_{f i} {)]}^{3 / 2}}\}}^{1 / 4}$ $R a_{\sqrt[]{s_{f i}}} = P r \frac{g β [\frac{t_{a p} (i) + t_{f i - o u t} (i)}{2} - t_{a i}] {(\sqrt[]{s_{f i}})}^{3}}{v^{2}}$ $β = 1 / [(\frac{t_{a p} (i) + t_{f i - o u t} (i)}{2} + t_{a i} + 273.15) / 2], f (P r) = 0.67 / [1 + {(0.5 / P r)}^{9 / 16}]^{4 / 9}$	含平行组合肋片的壳体与空气对流换热^[25]
Q_{conv, ap(fi)-ai}	$Q_{c o n v, a p (f i) - a i} (i) = A_{a p (f i)} (i) α_{c o n v, a p - a i} (i) (t_{a p} (i) - t_{a i} (i))$ $A_{a p (f i)} = \{\begin{matrix} Δ l L_{a p} - n_{f i} Δ l δ_{f i} & f o r f r o n t g r o u p s \\ 2 Δ l L_{a p} - 2 n_{f i} Δ l δ_{f i} & f o r r e a r g r o u p s \end{matrix}$	含有肋片的壳体平板部分与空气的对流换热
Q_{cond, ap-fi}	$Q_{c o n d, a p - f i} (i) = A_{f i} λ_{f i} \frac{(t_{a p} (i) - t_{f i - o u t} (i))}{δ_{f i}}$ $A_{f i} = \{\begin{matrix} n_{f i} Δ l δ_{f i} & f o r f r o n t g r o u p s \\ 2 n_{f i} Δ l δ_{f i} & f o r r e a r g r o u p s \end{matrix}$	壳体与肋片的导热
ΔP_m	$Δ P_{m} (i) = G_{r e}^{2} {[\frac{x_{r e} {(i)}^{2}}{ξ ρ_{r e - g} (i)} + \frac{(1 - x_{r e} {(i))}^{2}}{(1 - ξ) ρ_{r e - l} (i)}]}_{o u t} - G_{r e}^{2} {[\frac{x_{r e} {(i)}^{2}}{ξ ρ_{r e - g} (i)} + \frac{(1 - x_{r e} {(i))}^{2}}{(1 - ξ) ρ_{r e - l} (i)}]}_{i n}$ $ξ = 1 - \frac{2 (1 - x_{r e} {(i))}^{2}}{1 - 2 x_{r e} (i) + {[1 + 4 x_{r e} (i) (1 - x_{r e} (i)) (\frac{ρ_{r e - l} (i)}{ρ_{r e - g} (i)} - 1)]}^{0.5}}$	动量压降^[26]
ΔP_f	$Δ P_{f} (i) = - \frac{2 f (i) {G_{r e}}^{2} Δ l}{d_{c o - i n} ρ_{r e} (i)}$ , $f (i) = 0.0046 R e_{r e} {(i)}^{- 0.2}$	单相流流体摩擦压降^[27]
ΔP_f	$Δ P_{f} (i) = Δ P_{f - l} (i) ϕ {(i)}^{2}$ $ϕ {(i)}^{2} = 1 + \frac{C}{ψ} + \frac{1}{ψ^{2}}$ , $ψ^{2} = \frac{{(\frac{d P_{f}}{d z})}_{l}}{{(\frac{d P_{f}}{d z})}_{g}}$ ${(\frac{d P_{f}}{d z})}_{l} = \frac{2 f {(i)}_{l} {G_{r e}}^{2} (1 - x_{r e} {(i))}^{2}}{d_{c o - i n} ρ_{r e - l} (i)} {(\frac{d P_{f}}{d z})}_{g} = \frac{2 f {(i)}_{g} {G_{r e}}^{2} x_{r e} {(i)}^{2}}{d_{c o - i n} ρ_{r e - g} (i)}$ $C = \{\binom{\binom{0.39 R e_{r e - l}^{0.03} L a_{r e - g}^{0.1} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.35} R e_{r e - l} \geq 2000, R e_{r e - g} \geq 2000}{8.7 \times 10^{- 4} R e_{r e - l}^{0.17} L a_{r e - g}^{0.5} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.14} R e_{r e - l} \geq 2000, R e_{r e - g} < 2000}}{\binom{0.0015 R e_{r e - l}^{0.59} L a_{r e - g}^{0.19} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.36} R e_{r e - l} < 2000, R e_{r e - g} \geq 2000}{3.5 \times 10^{- 5} R e_{r e - l}^{0.44} L a_{r e - g}^{0.5} {(\frac{ρ_{r e - l}}{ρ_{r e - g}})}^{0.48} R e_{r e - l} < 2000, R e_{r e - g} < 2000}}$ $L a_{r e - g} = \frac{R {e_{r e - g}}^{2}}{W e_{r e - g}}, R e_{r e - g} = \frac{G_{r e} d_{c o - i n}}{μ_{r e - g}}, R e_{r e - l} = \frac{G_{r e} d_{c o - i n}}{μ_{r e - l}}, W e_{r e - g} = \frac{{G_{r e}}^{2} d_{c o - i n}}{ρ_{r e - g}}$	两相流流体摩擦压降^[28]

Table 3 Experimental results of the model

Parameters	Case 1	Case 2	Case 3
t_{re_in}/℃	52.2	57.5	66.8
P_{re_in}/kPa	2609.3	2826.3	3116.3
G_{re_total}/(kg/h)	43.9	49.6	56.7
t_ai	18.4	19.7	18.3
t_{wall_south}	18.2	18.7	18.0
t_{wall_east}	18.3	18.7	18.1
t_{wall_west}	18.2	18.7	18.2
t_{wall_north}	20.6	22.6	21.0
t_ceiling	18.3	18.9	18.2
t_floor	17.9	21.7	18.4
t_AUST	18.6	20.0	18.7

Fig.4 Comparison of numerical values with experimental values

Table 4 Ranges of operating parameters in numerical cases

工况组号	G_re/(kg/h)	P_re-in/kPa	t_re-in/℃	T_ai/℃	t_AUST/℃
a1	36~40	2726.1	49	18~22	16
a2	41~45	3062.8	54	18~22	16
a3	46~50	3431.3	59	18~22	16
b1	38	2662.3~2857.2	48~51	18~22	16
b2	42	2924.5~3133.9	52~55	18~22	16
b3	47	3206.2~3431.3	56~59	18~22	16
c1	38	2726.1	47~55	18~22	16
c2	42	3062.8	52~60	18~22	16
c3	47	3431.3	57~65	18~22	16
d1	38	2726.1	49	20	14~18
d2	42	3062.8	54	20	14~18
d3	47	3431.3	59	20	14~18

Fig.5 Thermal performance variations of the AHE under different refrigerant flow rates

Fig.6 Thermal performance variations of the AHE under different inlet pressures

Fig.7 Thermal performance variations of the AHE under different inlet temperatures

Fig.8 Thermal performance variations of the AHE under different indoor parameters

Fig.9 Each layer temperature changes of the AHE in 168 numerical conditions

Fig.10 Analysis of each layer temperature difference of the AHE in 168 numerical conditions

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