die Brühe
broth
A clear, savory liquid made by cooking meat, bones, fish, or vegetables in water, used as a base for soups or sauces.
Eine kräftige Hühnerbrühe hilft bei Erkältungen.
A strong chicken broth helps with colds.

broth / sludge
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broth
A clear, savory liquid made by cooking meat, bones, fish, or vegetables in water, used as a base for soups or sauces.
Eine kräftige Hühnerbrühe hilft bei Erkältungen.
A strong chicken broth helps with colds.

The meaning of 'Brühe' (broth/sludge) depends heavily on the context. In a cooking context, it is a soup base. In other contexts, the word is usually negative and describes a bad or dirty liquid.
Für die Soße brauchen wir eine gute Gemüsebrühe.
For the sauce, we need a good vegetable broth.
In the culinary sense, 'Brühe' (broth) is often used with verbs like 'kochen' (to cook), 'zubereiten' (to prepare), or 'ansetzen' (to start/prepare). Typical types are 'Hühnerbrühe' (chicken broth), 'Rinderbrühe' (beef broth), and 'Gemüsebrühe' (vegetable broth).
Ich koche die Nudeln direkt in der Gemüsebrühe, dann werden sie würziger.
I cook the noodles directly in the vegetable broth; that makes them more flavorful.
When 'Brühe' (brew/sludge) describes a bad or dirty liquid, the word is colloquial and derogatory. Adjectives like 'dünn' (thin), 'wässrig' (watery), or 'dreckig' (dirty) reinforce this negative meaning.
Nach dem Rohrbruch stand eine braune Brühe im ganzen Bad.
After the pipe burst, a brown sludge covered the whole bathroom.
The idiom is 'jemanden im eigenen Saft/in der eigenen Brühe schmoren lassen' (to let someone stew in their own juice/broth). It means leaving someone alone with the negative consequences of their actions and not helping them.
Er hat uns alle angelogen, jetzt lassen wir ihn in seiner eigenen Brühe schmoren.
He lied to all of us; now we're leaving him to stew in his own juice.
juice / power
'der Saft' and 'die Brühe' are both liquids, but they are made and used differently. 'Saft' is often a drink made from fruit or vegetables. In cooking, 'Saft' can also be the liquid that comes out of meat or vegetables. 'Brühe' is made by cooking meat, bones, fish, or vegetables in water. It is savory and is often used for soups or sauces. The words can be confused because both occur in the kitchen. But for a glass of orange juice, you do not say 'Brühe', and for a soup base, you normally do not say 'Saft'.

In der Küche kocht Clara eine Brühe aus Gemüse für die Suppe.
In the kitchen, Clara is making a vegetable broth for the soup.
Zum Frühstück trinkt Mehmet ein Glas Saft aus frischen Orangen.
For breakfast, Mehmet drinks a glass of juice made from fresh oranges.