A dairy farm needs to chill the milk from 39°C (extraction temperature is the same as the cow's body temperature) to at least 12°C for storage by using an existing 4-m long concentric-tube heat exchanger. The inner tube of the heat exchanger is 4 cm in diameter. The milk (density: 1035 kg/m³, specific heat: 3860 J/kg.K) flows into the heat exchanger at a rate of 270 liters per hour. On the cold side, there is a supply of water at 2°C at a rate of 0.23 kg/s. The estimated overall heat transfer coefficient of the heat exchanger is 1084.2 W/m².K. It was measured that the water comes out of the heat exchanger at 11°C. Is the milk indeed being cooled down to at least 12°C by heat exchanger as required by the farm? Looking at the temperatures obtained from your calculation, put an argument whether the heat exchanger is a parallel-flow or a counter-flow heat exchanger. Approximate specific heat of water is 4200 J/kg.K. Neglect radiation.
A dairy farm needs to chill the milk from 39°C (extraction temperature is the same as the cow's body temperature) to at least 12°C for storage by using an existing 4-m long concentric-tube heat exchanger. The inner tube of the heat exchanger is 4 cm in diameter. The milk (density: 1035 kg/m³, specific heat: 3860 J/kg.K) flows into the heat exchanger at a rate of 270 liters per hour. On the cold side, there is a supply of water at 2°C at a rate of 0.23 kg/s. The estimated overall heat transfer coefficient of the heat exchanger is 1084.2 W/m².K. It was measured that the water comes out of the heat exchanger at 11°C. Is the milk indeed being cooled down to at least 12°C by heat exchanger as required by the farm? Looking at the temperatures obtained from your calculation, put an argument whether the heat exchanger is a parallel-flow or a counter-flow heat exchanger. Approximate specific heat of water is 4200 J/kg.K. Neglect radiation.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Heat Exchangers
Heat exchangers are the types of equipment that are primarily employed to transfer the thermal energy from one fluid to another, provided that one of the fluids should be at a higher thermal energy content than the other fluid.
Heat Exchanger
The heat exchanger is a combination of two words ''Heat'' and ''Exchanger''. It is a mechanical device that is used to exchange heat energy between two fluids.
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Transcribed Image Text:A dairy farm needs to chill the milk from 39°C (extraction temperature is the same as the cow's body
temperature) to at least 12°C for storage by using an existing 4-m long concentric-tube heat exchanger.
The inner tube of the heat exchanger is 4 cm in diameter. The milk (density: 1035 kg/m³, specific heat:
3860 J/kg.K) flows into the heat exchanger at a rate of 270 liters per hour. On the cold side, there is
a supply of water at 2°C at a rate of 0.23 kg/s. The estimated overall heat transfer coefficient of the
heat exchanger is 1084.2 W/m².K. It was measured that the water comes out of the heat exchanger
at 11°C. Is the milk indeed being cooled down to at least 12°C by heat exchanger as required by the
farm? Looking at the temperatures obtained from your calculation, put an argument whether the heat
exchanger is a parallel-flow or a counter-flow heat exchanger. Approximate specific heat of water is
4200 J/kg.K. Neglect radiation.
![Effectiveness and NTU relationships for common heat exchangers
Effectiveness as a function of NTU and Cr:
Parallel flow:
Counter flow:
One shell and 2, 4, ... tube passes":
n shells and 2n, 4n, ... tube passes*:
Cross-flow both unmixed:
Cross-flow* Cmar on mixed side:
Cross-flow* Cmin on mixed side:
All heat exchangers with C, = 0:
NTU as a function of e and Cr:
Parallel flow:
Counter flow:
One shell and 2, 4, ... tube passest:
n shells and 2n, 4n, ... tube passes¹:
Cross-flow both unmixed:
Cross-flow" Cmaz on mixed side:
Cross-flow* Cmin on mixed side:
€ =
All heat exchangers with Cr = 0:
€ =
=
1 - exp[-NTU (1 + Cr)]
1+ Cr
1- exp[-NTU (1 - Cr)]
1 - C, exp[-NTU (1 — Cr)]'
NTU
1+ NTU'
when C₂ = 1
€1 = 21+ C₂ + (1+C²) ¹1/²
€ =
1
€ =
= [(²₁₁ ²¹+ )" - ¹] [(²¹²¹)" - c]'
1-
€1
€1
Use ₁ and NTU₁ from Eqn. 4
€ = 1- exp
F
NTU=
(1 - exp{-C, [1 - exp(-NTU)]})
NTU=
€ = 1 - exp[-C₂¹ (1 - exp[-CrNTU])]
€ = 1- exp(- NTU)
=
[(+)₁ NTU0.22 [exp(-C, NTU0.78) - :
1+ exp-NTU₁ (1+ -C²) ¹/2
1- exp -NTU₁ (1+C²) ¹/2]
C₂²-1¹n (6--11),
In
Cr
In [1 - € (1 + Cr)]
1+ Cr
1 when C₁ = 1
€
NTU₁ = (1 + C²)-¹/2 In
€1 =
when Cr < 1
F-1
F-Cr
Solve Eqn. 6 numerically
NTU - In
NTU=-
NTU = nx NTU₁, use NTU₁ is from Eqn. 13 with €₁ as
1/n
= (Cr-1) ¹/
F =
1
Cr
ln (1-ɛ)
when Cr < 1
(E = 1).
- In [1 + / - In (1 - eCr)]
E =
2 − (1+Cr)
(1+C2) ¹1/2
In [Cr In (1 e) + 1]
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(16)
NTU=
(17)
1: If it is only a one-shell heat exchanger, € = ₁ and NTU = NTU₁. If there are n shells, use the ₁ and NTU₁ calculated based on one
shell formulation and calculate final and NTU for the entire heat exchanger. Also note, number of tubes does not directly feature in these
equations, they come in via the total area calculation.
*: Single pass crossflow heat exchangers
(13)
(14)
(15)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20000a63-5a8a-4b5e-adbd-85ecbf6c298b%2Fec59b577-64de-44c7-9120-f1024f519d08%2F4oqyr8g_processed.png&w=3840&q=75)
Transcribed Image Text:Effectiveness and NTU relationships for common heat exchangers
Effectiveness as a function of NTU and Cr:
Parallel flow:
Counter flow:
One shell and 2, 4, ... tube passes":
n shells and 2n, 4n, ... tube passes*:
Cross-flow both unmixed:
Cross-flow* Cmar on mixed side:
Cross-flow* Cmin on mixed side:
All heat exchangers with C, = 0:
NTU as a function of e and Cr:
Parallel flow:
Counter flow:
One shell and 2, 4, ... tube passest:
n shells and 2n, 4n, ... tube passes¹:
Cross-flow both unmixed:
Cross-flow" Cmaz on mixed side:
Cross-flow* Cmin on mixed side:
€ =
All heat exchangers with Cr = 0:
€ =
=
1 - exp[-NTU (1 + Cr)]
1+ Cr
1- exp[-NTU (1 - Cr)]
1 - C, exp[-NTU (1 — Cr)]'
NTU
1+ NTU'
when C₂ = 1
€1 = 21+ C₂ + (1+C²) ¹1/²
€ =
1
€ =
= [(²₁₁ ²¹+ )" - ¹] [(²¹²¹)" - c]'
1-
€1
€1
Use ₁ and NTU₁ from Eqn. 4
€ = 1- exp
F
NTU=
(1 - exp{-C, [1 - exp(-NTU)]})
NTU=
€ = 1 - exp[-C₂¹ (1 - exp[-CrNTU])]
€ = 1- exp(- NTU)
=
[(+)₁ NTU0.22 [exp(-C, NTU0.78) - :
1+ exp-NTU₁ (1+ -C²) ¹/2
1- exp -NTU₁ (1+C²) ¹/2]
C₂²-1¹n (6--11),
In
Cr
In [1 - € (1 + Cr)]
1+ Cr
1 when C₁ = 1
€
NTU₁ = (1 + C²)-¹/2 In
€1 =
when Cr < 1
F-1
F-Cr
Solve Eqn. 6 numerically
NTU - In
NTU=-
NTU = nx NTU₁, use NTU₁ is from Eqn. 13 with €₁ as
1/n
= (Cr-1) ¹/
F =
1
Cr
ln (1-ɛ)
when Cr < 1
(E = 1).
- In [1 + / - In (1 - eCr)]
E =
2 − (1+Cr)
(1+C2) ¹1/2
In [Cr In (1 e) + 1]
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(16)
NTU=
(17)
1: If it is only a one-shell heat exchanger, € = ₁ and NTU = NTU₁. If there are n shells, use the ₁ and NTU₁ calculated based on one
shell formulation and calculate final and NTU for the entire heat exchanger. Also note, number of tubes does not directly feature in these
equations, they come in via the total area calculation.
*: Single pass crossflow heat exchangers
(13)
(14)
(15)
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