Engineering, 25.04.2020 04:26 liyahlanderson2232

Cold water (cp = 4180 J/kg·K) leading to a shower enters a thin-walled double-pipe counterflow heat exchanger at 15°C at a rate of 0.25 kg/s and is heated to 45°C by hot water (cp = 4190 J/kg·K) that enters at 100°C at a rate of 3 kg/s. If the overall heat transfer coefficient is 950 W/m2·K, determine the rate of heat transfer and the heat transfer surface area of the heat exchanger using the ε–NTU method. Answers: 31.35 kW, 0.482 m2

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Answer from: travisjpatterson

The rate of heat transfer is $H = 31.35\ kW$

The heat transfer surface area is A_s = 0.4818 m^2

Explanation:

From the question we are told that

The specific heat of water is $cp = 4180 \ J/kg \cdot K$

The temperature of cold water is T_c = 15^o C

The rate of cold the flow is $\r m = 0.25 kg/s$

The temperature of the heated water T_h = 45 ^oC

The specific heat of hot water is $c_p__{H}} = 4190 J/kg \cdot K$

The temperature of the hot water is T_H = 100^oC

The rate of hot the flow is $\r m_H = 3 kg/s$

The heat transfer coefficient is $U = 950 W/m^2 \cdot K,$

From the $\epsilon -NTU$ method we have that

$C_h = \r m_H c_p__{H}}$

Where C_h is the heat capacity rate of hot water

Substituting the value

C_h = 3 * (4190)

$= 12,5700\ W/^oC$

Also

$C_c = \r m c_p$

Where C_c is the heat capacity rate of cold water

C_c = 0.25 * 4180

$= 1045 \ W / ^oC$

The maximum heat capacity C_h and the minimum heat capacity is C_c

The maximum heat transfer is

$H_{max} = C_c (T_H - T_c)$

Substituting values

$H_{max} = (1045)(100- 15)$

$= 88,825\ W$

The actual heat transfer is mathematically evaluated as

H = C_c (T_h - T_c)

Substituting values

H = 1045 (45 - 15 )

$H = 31350 \ W$

$H = 31.35\ kW$

The effectiveness of the heat exchanger is mathematically evaluated as

$\epsilon = \frac{H}{H_{max}}$

Substituting values

$\epsilon = \frac{31350}{88,825}$

= 0.35

The NTU of the heat exchanger is mathematically represented as

$NTU = \frac{1}{C-1} ln [\frac{\epsilon - 1}{\epsilon C -1} ]$

Where C is the ratio of the minimum to the maximum heat capacity which is mathematically represented as

$C = \frac{C_c}{C_h}$

Substituting values

$C = \frac{1045}{12,570}$

= 0.083

Substituting values in to the equation for NTU

$NTU = \frac{1}{0.083 -1} ln[\frac{0.35 - 1}{0.35 * (0.083 - 1)} ]$

= 0.438

Generally the heat transfer surface area can be mathematically represented as

$A_s = \frac{NTU C_c}{U}$

Substituting values

$A_s = \frac{(0.438) (1045)}{950}$

A_s = 0.4818 m^2

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Answer from: Quest

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